Introduction
Wilson’s lattice gauge theory [77 ] assigns an element Q e Q_e Q e G ⊂ U ( N ) G \subset U(N) G ⊂ U ( N ) e e e Z 4 \mathbb{Z}^4 Z 4 exp ( N β W ∑ p Re Tr Q p ) \exp(N\beta_{\mathrm{W}}\sum_p \operatorname{Re}\operatorname{Tr} Q_p) exp ( N β W ∑ p Re Tr Q p ) Q p Q_p Q p p p p 1 ∣ β ∣ |\beta| ∣ β ∣ [64 , 68 ] .2 [47 ] begins where strong coupling ends and Bałaban’s weak-coupling analysis [5 , 6 , 7 ] takes over.
Two methods give explicit ranges. The Bakry–Émery method [3 , 4 ] of Shen, Zhu and Zhu [69 ] bounds the Hessian of the action; in [14 ] we showed that the sharp bound is 4 d = 16 4d=16 4 d = 16 ∣ β ∣ < 1 / 32 |\beta|<1/32 ∣ β ∣ < 1/32 [29 , 30 ] 3 4 [14 ] (Theorem 125 in Appendix § 1 ), and every link has eighteen partners, which gives β W < 2 / 9 \beta_{\mathrm{W}}<2/9 β W < 2/9
Both limits come from the same count. A link of Z 4 \mathbb{Z}^{4} Z 4 [26 ] .5 Lemma 9 ), and both methods improve.
Theorem 1 (SU(2)). Let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) 0 ≤ β W < 1 3 0\leq\beta_{\mathrm{W}}<\frac{1}{3} 0 ≤ β W < 3 1 O O O W Λ ( O ) W_{\Lambda}(O) W Λ ( O ) Z 4 \mathbb{Z}^{4} Z 4 ω ( O ) \omega(O) ω ( O ) ω \omega ω 6 − log ( 3 β W ) -\log(3\beta_{\mathrm{W}}) − log ( 3 β W )
m ≥ − log ( 3 β W ) > 0. m\geq-\log(3\beta_{\mathrm{W}})>0. m ≥ − log ( 3 β W ) > 0. For − 1 3 < β W < 0 -\frac{1}{3}<\beta_{\mathrm{W}}<0 − 3 1 < β W < 0
The step behind Theorem 1 is short once the line field is in place. The number 1 3 \frac{1}{3} 3 1
Theorem 2 (Ceiling). Let F ′ F' F ′ Z 4 \mathbb{Z}^{4} Z 4 F ′ F' F ′ ∣ β ∣ |\beta| ∣ β ∣ β W ≥ 1 3 \beta_{\mathrm{W}}\geq\frac{1}{3} β W ≥ 3 1 β W = 1 3 \beta_{\mathrm{W}}=\frac{1}{3} β W = 3 1
The ceiling concerns coefficients between single links. Take instead as a block the three free links of one plaquette. The free links of the line field split into such triples with period two (Lemma 12), the coupling inside the block plaquette is no longer paid for, and each link keeps ten outside partners. The price is the sensitivity of a law on three spheres instead of one, which we bound analytically by integrating out the middle link of the block.
Theorem 3 (SU(2) with plaquette blocks). Let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) 0 ≤ β W ≤ 0.345 0\leq\beta_{\mathrm{W}}\leq0.345 0 ≤ β W ≤ 0.345 α ( β W ) \alpha(\beta_{\mathrm{W}}) α ( β W ) (4) is increasing, α ( 1 3 ) ≤ 0.9561 \alpha(\frac{1}{3})\leq0.9561 α ( 3 1 ) ≤ 0.9561 α ( 0.345 ) ≤ 0.9973 \alpha(0.345)\leq0.9973 α ( 0.345 ) ≤ 0.9973 Theorem 1 hold with decay at every rate below − 1 2 log α ( β W ) -\frac{1}{2}\log\alpha(\beta_{\mathrm{W}}) − 2 1 log α ( β W )
m ≥ − 1 2 log α ( β W ) > 0 ; m\geq-\frac{1}{2}\log\alpha(\beta_{\mathrm{W}})>0; m ≥ − 2 1 log α ( β W ) > 0 ; at β W = 1 3 \beta_{\mathrm{W}}=\frac{1}{3} β W = 3 1 m ≥ 0.0224 m\geq0.0224 m ≥ 0.0224 − 0.345 ≤ β W < 0 -0.345\leq\beta_{\mathrm{W}}<0 − 0.345 ≤ β W < 0
At small coupling the rate of Theorem 1 is the better one, so both bounds stand side by side; Theorem 3 matters past β W = 1 3 \beta_{\mathrm{W}}=\frac{1}{3} β W = 3 1 Theorem 1 has closed.
The curvature method gains as well, and for every N N N Hess S 0 \operatorname{Hess}S_{0} Hess S 0 ∑ p Re Tr Q p \sum_{p}\operatorname{Re}\operatorname{Tr}Q_{p} ∑ p Re Tr Q p G E G^{E} G E F F F
Theorem 4 (Curvature in the gauge). For every N N N every field Q Q Q and every tangent vector v = ( X e Q e ) v=(X_eQ_e) v = ( X e Q e ) with X e = 0 X_e=0 X e = 0 for e ∈ F e\in F e ∈ F
∣ Hess S 0 ( v , v ) ∣ ≤ 29 2 ∣ v ∣ 2 , \lvert\operatorname{Hess}S_0(v,v)\rvert\leq\frac{29}{2}\lvert v\rvert^2, ∣ Hess S 0 ( v , v )∣ ≤ 2 29 ∣ v ∣ 2 , against the sharp constant 16 without gauge fixing. Consequently, for G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) ∣ β ∣ < 1 29 \lvert\beta\rvert<\frac{1}{29} ∣ β ∣ < 29 1 G = S O ( N ) G=\mathrm{SO}(N) G = SO ( N ) N ≥ 3 N\geq3 N ≥ 3 ∣ β ∣ < N − 2 58 N \lvert\beta\rvert<\frac{N-2}{58N} ∣ β ∣ < 58 N N − 2 Theorem 1 hold with the rate and gap of Theorem 5 . For S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) β W < 9 29 = 0.310 \beta_W<\frac{9}{29}=0.310 β W < 29 9 = 0.310
The constant 29 2 \frac{29}{2} 2 29 U ( N ) \mathrm{U}(N) U ( N ) ρ : G → U ( N ) \rho:G\to\mathrm{U}(N) ρ : G → U ( N ) G G G exp ( N β ∑ p Re Tr ρ ( Q p ) ) \exp(N\beta\sum_p\operatorname{Re}\operatorname{Tr}\rho(Q_p)) exp ( N β ∑ p Re Tr ρ ( Q p )) κ G \kappa_G κ G 7 G G G ⟨ X , Y ⟩ = − Tr ( ρ ( X ) ρ ( Y ) ) \langle X,Y\rangle=-\operatorname{Tr}(\rho(X)\rho(Y)) ⟨ X , Y ⟩ = − Tr ( ρ ( X ) ρ ( Y )) G G G ρ ( G ) ⊂ U ( N ) \rho(G)\subset\mathrm{U}(N) ρ ( G ) ⊂ U ( N )
Theorem 5 (Every compact simple group). Suppose that the Lie algebra of G G G is semisimple, so that κ G > 0 \kappa_G>0 κ G > 0 and that
N ∣ β ∣ < 2 29 κ G . N\lvert\beta\rvert<\frac{2}{29}\kappa_G. N ∣ β ∣ < 29 2 κ G . Then the conclusions of Theorem 1 hold, for β < 0 \beta<0 β < 0
m G ( β ) = arccosh ( 1 + 1 12 ( κ G N ∣ β ∣ − 29 2 ) ) > 0 m_G(\beta)=\operatorname{arccosh}\left(1+\frac{1}{12}\left(\frac{\kappa_G}{N\lvert\beta\rvert}-\frac{29}{2}\right)\right)>0 m G ( β ) = arccosh ( 1 + 12 1 ( N ∣ β ∣ κ G − 2 29 ) ) > 0 and a transfer-matrix gap m ≥ m G ( β ) m\geq m_G(\beta) m ≥ m G ( β ) G G G κ G = h ∨ / ( 2 ℓ ρ ) \kappa_G=h^\vee/(2\ell_\rho) κ G = h ∨ / ( 2 ℓ ρ ) h ∨ h^\vee h ∨ ℓ ρ \ell_\rho ℓ ρ ρ \rho ρ Table I lists the values.
For S U ( N ) \mathrm{SU}(N) SU ( N ) κ G = N / 2 \kappa_G=N/2 κ G = N /2 S O ( N ) \mathrm{SO}(N) SO ( N ) κ G = ( N − 2 ) / 4 \kappa_G=(N-2)/4 κ G = ( N − 2 ) /4 Theorem 5 contains the ranges of Theorem 4 and supplies their rate.8 ∣ β ∣ \lvert\beta\rvert ∣ β ∣ [64 Thm. 3.6] ; what is new is an explicit range for every group, past the range N ∣ β ∣ < κ G / 16 N\lvert\beta\rvert<\kappa_G/16 N ∣ β ∣ < κ G /16 β → 0 \beta\to0 β → 0
log κ G 6 N ∣ β ∣ . \log\frac{\kappa_G}{6N\lvert\beta\rvert}. log 6 N ∣ β ∣ κ G . No Bakry–Émery argument for the ungauged measure can pass ∣ β ∣ = 1 32 \lvert\beta\rvert=\frac{1}{32} ∣ β ∣ = 32 1 [14 Thm. C] ; the gauge fixing changes the measure, not the metric, which is why it can.
Table I. Least Ricci curvature κ G = h ∨ / ( 2 ℓ ρ ) \kappa_G=h^\vee/(2\ell_\rho) κ G = h ∨ / ( 2 ℓ ρ ) ρ \rho ρ N N N Theorem 5 for the conventional inverse coupling β W = N 2 β \beta_W=N^2\beta β W = N 2 β
For S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) β W < 4 29 \beta_W<\frac{4}{29} β W < 29 4 Theorem 1 , and no curvature argument for the gauge-fixed S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) β W = 1 3 \beta_W=\frac{1}{3} β W = 3 1 e e e − 1 -1 − 1 1 1 1 e e e − 1 -1 − 1 Q e Q_e Q e − β W 2 ∑ p Re Tr Q p -\frac{\beta_W}{2}\sum_p\operatorname{Re}\operatorname{Tr}Q_p − 2 β W ∑ p Re Tr Q p − 6 β W -6\beta_W − 6 β W S 3 S^3 S 3 2 2 2 β W > 1 3 \beta_W>\frac{1}{3} β W > 3 1 S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
Theorem 6 (S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) Let G = S U ( 3 ) G=\mathrm{SU}(3) G = SU ( 3 ) and 0 ≤ β W ≤ 3 4 0\leq\beta_W\leq\frac{3}{4} 0 ≤ β W ≤ 4 3 Then the conclusions of Theorem 1 hold, with decay at every rate below − log ( 1.33319 β W ) -\log(1.33319\beta_W) − log ( 1.33319 β W ) and a transfer-matrix gap
m ≥ − log ( 1.33319 β W ) > 0. m\geq-\log(1.33319\beta_W)>0. m ≥ − log ( 1.33319 β W ) > 0. The proof of Theorem 6 bounds the Kantorovich modulus of the one-link measure ν b ( d U ) ∝ e Re Tr ( U b ) d U \nu_b(dU)\propto e^{\operatorname{Re}\operatorname{Tr}(Ub)}dU ν b ( d U ) ∝ e Re Tr ( U b ) d U S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) b b b ν b \nu_b ν b b b b ν b \nu_b ν b Theorem 3 work for S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) Λ ( 2 β W ) \Lambda(2\beta_W) Λ ( 2 β W ) 2 r Λ ( r ) < 1 2r\Lambda(r)<1 2 r Λ ( r ) < 1 Theorem 6 they give a much larger gap, and they go past it.
Theorem 7 (S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) Let G = S U ( 3 ) G=\mathrm{SU}(3) G = SU ( 3 ) and 0 ≤ β W ≤ 19 25 = 0.76 0\leq\beta_W\leq\frac{19}{25}=0.76 0 ≤ β W ≤ 25 19 = 0.76 Then α 3 ( β W ) = 1 \alpha_3(\beta_W)=1 α 3 ( β W ) = 1 for the number of (15), with α 3 ( 0.72 ) ≤ 0.9182 \alpha_3(0.72)\leq0.9182 α 3 ( 0.72 ) ≤ 0.9182 α 3 ( 3 4 ) ≤ 0.9653 \alpha_3(\frac{3}{4})\leq0.9653 α 3 ( 4 3 ) ≤ 0.9653 and α 3 ( 0.76 ) ≤ 0.99950 \alpha_3(0.76)\leq0.99950 α 3 ( 0.76 ) ≤ 0.99950 The conclusions of Theorem 1 hold with decay at every rate below − 1 2 log α 3 ( β W ) -\frac{1}{2}\log\alpha_3(\beta_W) − 2 1 log α 3 ( β W ) and a transfer-matrix gap
m ≥ − 1 2 log α 3 ( β W ) > 0 ; m\geq-\frac{1}{2}\log\alpha_3(\beta_W)>0; m ≥ − 2 1 log α 3 ( β W ) > 0 ; Figure 1. The gap m G ( β ) m_G(\beta) m G ( β ) Theorem 5 as a function of β W = N 2 β \beta_W=N^2\beta β W = N 2 β Table I . Each curve ends at the edge β W = 2 N κ G / 29 \beta_W=2N\kappa_G/29 β W = 2 N κ G /29 β W → 0 \beta_W\to0 β W → 0 log ( κ G / ( 6 N ∣ β ∣ ) ) \log(\kappa_G/(6N|\beta|)) log ( κ G / ( 6 N ∣ β ∣ )) N κ G N\kappa_G N κ G m G m_G m G N ∣ β ∣ / κ G N|\beta|/\kappa_G N ∣ β ∣/ κ G
Table 2. Explicit ranges of uniqueness, decay and a transfer-matrix gap for four-dimensional lattice Yang–Mills theory. The first three rows hold for the ungauged measure; the last three are the results of this paper.
at β W = 3 4 \beta_W=\frac{3}{4} β W = 4 3 m ≥ 0.0177 m\ge0.0177 m ≥ 0.0177 1.1 ⋅ 10 − 4 1.1\cdot10^{-4} 1.1 ⋅ 1 0 − 4 Theorem 6 , and at β W = 0.76 \beta_W=0.76 β W = 0.76 Theorem 6 no longer applies, m ≥ 2.5 ⋅ 10 − 4 m\ge2.5\cdot10^{-4} m ≥ 2.5 ⋅ 1 0 − 4
All seven theorems are statements at fixed lattice spacing. They do not touch the continuum limit, the crossover to Bałaban’s regime, or a gap in physical units, which together are the Clay problem; § 2.6 says where each method stops and what remains of the four open problems of [14 ] . The obstruction proved in [15 ] , that a lattice gap uniform in lattice units forces trivial continuum two-point functions, applies to all of them.
The continuum step. Sec. X goes past that obstruction in one respect, through three implications, each proved.
(I1) A Dobrushin–Shlosman condition on one cube of fixed physical size, uniform along the continuum tuning, gives a mass gap in physical units for every continuum limit (Theorems 43 and 45 ), and so does exponential decay with constants of the renormalised correlations of a spanning set of observables (Theorem 46 ); this holds also for observables smoothed by the spatial gradient flow (Proposition 47 ).
(I2) The flowed observables, renormalised by one condition at a single physical separation, have reflection positive continuum limits along subsequences, and such a limit is nontrivial exactly when one ratio of lattice two-point functions stays away from zero (Theorem 48 ); that ratio is bounded above and below by ratios of correlations of one slice field at fixed physical separations (Proposition 51).
(I3) Tree bounds on the truncated correlations give all the higher Schwinger functions as reflection positive distributions, imply the ratio bound, and carry the conditional mass gap to every polynomial in the smeared fields (Theorem 56); for a smooth packet they give the ratio with an explicit rate and a reference vector of finite energy (Proposition 60).
What is open. The inputs of (I1)–(I3), namely the cube condition along the tuning, the slice ratio in four dimensions and the tree bounds, are not proved, and so the nontrivial existence of the continuum limit and the gap in physical units remain open. Figure 3 shows how the proved statements, the implications and the open inputs fit together.
Figure 3. The logical structure of the paper. Green: theorems at fixed lattice spacing (P1 the gap for every compact simple group, Theorem 5 , P2 the plaquette blocks for SU(2) and SU(3), P3 the cube condition at strong coupling). Blue: the proved implications (I1)–(I3) of the continuum step. Red: their inputs, none of which is proved along the continuum tuning. Each arrow joins an open input to the implication that consumes it; no implication feeds its own input.
Sec. XI measures the open inputs. The cube condition holds at strong coupling, fails at every fixed cube size at weak coupling, forces the gauge-fixed variables to be disordered on the scale of the cube, and cannot be reached from estimates on single lines of links; for SU(N) it needs cubes of side at least of order ( β W / log β W ) 1 / 4 (\beta_W/\log\beta_W)^{1/4} ( β W / log β W ) 1/4 Proposition 59 ). The slice ratio holds in two dimensions, where it is the exact solution of lattice gauge theory (Remark 53), and fails for a field whose mass is fixed in lattice units; for the free field, conditioning an energy packet on averages over blocks leaves a fixed fraction of its ordinary variance at every lattice spacing, so a reduction to block averages has to be measured in the reflected norm (Proposition 54).
Figure 2. Ranges of Table II. The height of each bar is the largest inverse coupling β W \beta_W β W Theorem 2 and the orange bar, plaquette blocks, passes it (Theorem 3 ); for SU(3) the red bar is Theorem 6 and the orange bar, plaquette blocks, gives a larger gap there and passes it (Theorem 7 ). Legend: ungauged curvature (Shen, Zhu, Zhu); ungauged curvature, sharp; ungauged Dobrushin, sharp quarter; forest gauge, curvature; forest gauge, single-link Dobrushin; forest gauge, plaquette blocks.
Block averages. Sec. XII poses the cube condition for block averages in a gauge whose fixed links are trees inside blocks of side two. Its quadratic approximation then holds at every coupling once the cube is large (Theorem 88), and integrating out one scale reduces it to a cube condition for the averages, which carry at least the coarse quadratic form at every scale (Corollary 91 and Proposition 89); the averages along the lines of the forest gauge, by contrast, leave one class of lines invisible and fail at every size (Proposition 82). For the group-valued links, the straight path between block roots gives a coarse link whose fibres satisfy a coercivity bound for every compact group and every field (Theorem 94 ), but this map does not compose across scales (Proposition 97), and for every compact connected group no continuous covariant block map treats the positions of a block alike (Proposition 98). A cut-off of Bałaban’s averaging operation is continuous, covariant and coercive for every field and agrees with the block averages to first order; it breaks the symmetry only at large fields, as it must, and its constants are those of one step (Theorem 100 ). On fields with values in a maximal torus whose curvature stays small at every scale it composes, with the energy nonincreasing and the thin paths contracted (Theorem 102 ), and for every compact group it composes on fields whose curvatures at the successive scales have a small sum (Theorem 106 ). Under the SU(2) Wilson measure the plaquettes that are not small lie in clusters whose physical size tends to zero in probability, and the cut-off acts only next to them (Theorem 108 , Corollaries 109 and 110); such plaquettes are nevertheless present along the one-loop tuning (Proposition 111). Whether some block map is coercive for all fields and composes across scales, through these clusters, is open.
Plan. § 1.1 constructs the line field. § 2 shows that a Dobrushin bound for the gauge-fixed specification gives the full conclusion of Theorem 1 . Secs. IV, § 2.2 , VI, VII and § 2.5 prove Theorems 1 –7 , and § 2.6 measures what is left. Sec. X proves (11)–(13), Sec. XI states and measures their open inputs, and Sec. XII poses the cube condition for block averages, composes the cut-off average across scales at small fields and locates the large fields under the Wilson measure. The appendices contain the residue table of the line field, the comparison lemma and its version for blocks, the blocks of the local Hessian inequality, the details of the certified computation, the reconstruction of the transfer matrix with the equivalence of a gap and exponential clustering (Appendix § 2.4.6 ), the facts on spectral measures and on the character expansion that the continuum step uses (Appendices § 2.4.7 and § 3.15 ), and a complete proof of the sharp quarter for the single-link law of SU(2) (Appendices § 1 –K), so that the paper does not depend on our earlier one for any step.
The line field
Write e 0 , … , e 3 e_0,\ldots,e_3 e 0 , … , e 3 Z 4 \mathbb{Z}^4 Z 4 ( x , μ ) (x,\mu) ( x , μ ) x x x x + e μ x+e_\mu x + e μ direction field is a map σ : Z 4 → { 0 , 1 , 2 , 3 } \sigma:\mathbb{Z}^4\to\{0,1,2,3\} σ : Z 4 → { 0 , 1 , 2 , 3 } F = { ( x , σ ( x ) ) : x ∈ Z 4 } F=\{(x,\sigma(x)):x\in\mathbb{Z}^4\} F = {( x , σ ( x )) : x ∈ Z 4 } π x ( μ ) = σ ( x + e μ ) \pi_x(\mu)=\sigma(x+e_\mu) π x ( μ ) = σ ( x + e μ ) x x x { μ , ν } \{\mu,\nu\} { μ , ν } ( x , μ ) (x,\mu) ( x , μ ) ( x + e μ , ν ) (x+e_\mu,\nu) ( x + e μ , ν ) ( x + e ν , μ ) (x+e_\nu,\mu) ( x + e ν , μ ) ( x , ν ) (x,\nu) ( x , ν )
[ σ ( x ) ∈ { μ , ν } ] + [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] (1) [\sigma(x)\in\{\mu,\nu\}]+[\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu] \tag*{(1)} [ σ ( x ) ∈ { μ , ν }] + [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] (1) links of F F F
Lemma 8. Every plaquette of Z 4 \mathbb{Z}^4 Z 4 F F F x x x π x \pi_x π x σ ( x ) \sigma(x) σ ( x )
Proof. Fix x x x s = σ ( x ) s=\sigma(x) s = σ ( x ) R R R { s , ν } \{s,\nu\} { s , ν } 1 + [ π x ( s ) = ν ] + [ π x ( ν ) = s ] 1+[\pi_x(s)=\nu]+[\pi_x(\nu)=s] 1 + [ π x ( s ) = ν ] + [ π x ( ν ) = s ] ν ∈ R \nu\in R ν ∈ R π x ( s ) = s \pi_x(s)=s π x ( s ) = s π x ( ν ) ≠ s \pi_x(\nu)\ne s π x ( ν ) = s ν ∈ R \nu\in R ν ∈ R π x \pi_x π x s s s R R R R R R { μ , ν } ⊂ R \{\mu,\nu\}\subset R { μ , ν } ⊂ R [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] [\pi_x(\mu)=\nu]+[\pi_x(\nu)=\mu] [ π x ( μ ) = ν ] + [ π x ( ν ) = μ ] μ → π x ( μ ) \mu\to\pi_x(\mu) μ → π x ( μ ) μ ∈ R \mu\in R μ ∈ R π x ( μ ) ∈ R ∖ { μ } \pi_x(\mu)\in R\setminus\{\mu\} π x ( μ ) ∈ R ∖ { μ } R R R R R R □ \square □
From now on σ \sigma σ Table III of Appendix A, read from x x x π x \pi_x π x π x \pi_x π x σ ( x ) \sigma(x) σ ( x ) F F F 9
Table III. The line field: x x x x 0 x 1 x 2 x 3 x_0x_1x_2x_3 x 0 x 1 x 2 x 3 σ ( x ) \sigma(x) σ ( x ) σ ( x + e μ ) \sigma(x+e_\mu) σ ( x + e μ ) μ = 0 , 1 , 2 , 3 \mu=0,1,2,3 μ = 0 , 1 , 2 , 3
Lemma 9. (i) F F F F F F σ ( x ) \sigma(x) σ ( x )
(ii) For every box Λ \Lambda Λ F ∩ E ( Λ ) F \cap E(\Lambda) F ∩ E ( Λ )
(iii) A link e = ( x , μ ) ∉ F e=(x,\mu)\notin F e = ( x , μ ) ∈ / F F F F e e e { μ , σ ( x ) } \{\mu,\sigma(x)\} { μ , σ ( x )} { μ , π x ( μ ) } \{\mu,\pi_x(\mu)\} { μ , π x ( μ )} { μ , ρ } \{\mu,\rho\} { μ , ρ } F F F e e e
(iv) Every link outside F F F F F F
Proof. (i) Since π x \pi_x π x σ ( x ) \sigma(x) σ ( x ) σ ( x + e σ ( x ) ) = σ ( x ) \sigma(x+e_{\sigma(x)})=\sigma(x) σ ( x + e σ ( x ) ) = σ ( x ) σ \sigma σ x − e μ ≡ x + e μ x-e_\mu\equiv x+e_\mu x − e μ ≡ x + e μ σ ( x − e σ ( x ) ) = σ ( x ) \sigma(x-e_{\sigma(x)})=\sigma(x) σ ( x − e σ ( x ) ) = σ ( x ) ( x − e σ ( x ) , σ ( x ) ) (x-e_{\sigma(x)},\sigma(x)) ( x − e σ ( x ) , σ ( x )) F F F ( x − e ν , ν ) ∈ F (x-e_\nu,\nu)\in F ( x − e ν , ν ) ∈ F σ ( x − e ν ) = ν \sigma(x-e_\nu)=\nu σ ( x − e ν ) = ν x − e ν x-e_\nu x − e ν σ ( x ) = ν \sigma(x)=\nu σ ( x ) = ν
(ii) A line meets a box in a segment, and distinct lines are disjoint.
(iii) The plaquettes of e e e ν ≠ μ \nu\ne\mu ν = μ x x x x − e ν x-e_\nu x − e ν { μ , ν } \{\mu,\nu\} { μ , ν } F F F e e e F F F x x x x + e μ x+e_\mu x + e μ x x x s = σ ( x ) ≠ μ s=\sigma(x)\ne\mu s = σ ( x ) = μ x x x x − e s x-e_s x − e s { μ , s } \{\mu,s\} { μ , s } x + e μ x+e_\mu x + e μ t = π x ( μ ) t=\pi_x(\mu) t = π x ( μ ) R ∖ { μ } \mathbb{R}\setminus\{\mu\} R ∖ { μ } π x \pi_x π x R R R x x x x − e t x-e_t x − e t { μ , t } \{\mu,t\} { μ , t } s ≠ t s\ne t s = t F F F { μ , ρ } \{\mu,\rho\} { μ , ρ } F F F μ \mu μ ρ \rho ρ e e e ρ \rho ρ e e e
(iv) Each of the six plaquettes contains e e e F F F
Gauge fixing and the free-boundary state
Let G ⊂ U ( N ) G\subset U(N) G ⊂ U ( N ) d U dU d U Λ \Lambda Λ
W Λ ( d Q ) = 1 Z Λ exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p ) ∏ e ∈ E ( Λ ) d Q e . W_\Lambda(dQ)=\frac{1}{Z_\Lambda}\exp\left(N\beta\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p\right)\prod_{e\in E(\Lambda)}dQ_e. W Λ ( d Q ) = Z Λ 1 exp N β p ∈ P ( Λ ) ∑ Re Tr Q p e ∈ E ( Λ ) ∏ d Q e . equivalent to σ \sigma σ 2 ( d − 1 ) 2(d-1) 2 ( d − 1 ) d ( d − 1 ) / 2 d(d-1)/2 d ( d − 1 ) /2 2 ( d − 1 ) = d ( d − 1 ) / 2 2(d-1)=d(d-1)/2 2 ( d − 1 ) = d ( d − 1 ) /2 d = 4 d=4 d = 4
The measure is invariant under the gauge transformations Q ( x , y ) ↦ g x Q ( x , y ) g y − 1 Q_{(x,y)}\mapsto g_xQ_{(x,y)}g_y^{-1} Q ( x , y ) ↦ g x Q ( x , y ) g y − 1 g ∈ G Λ g\in G^\Lambda g ∈ G Λ F F F A A A O O O O F O^F O F O O O F F F
Lemma 10 (Gauge fixing). Let μ Λ F \mu_\Lambda^F μ Λ F G A ∩ E ( Λ ) G^{A\cap E(\Lambda)} G A ∩ E ( Λ ) exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p F ) \exp(N\beta\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p^F) exp ( N β ∑ p ∈ P ( Λ ) Re Tr Q p F ) O O O G E ( Λ ) G^{E(\Lambda)} G E ( Λ )
W Λ ( O ) = μ Λ F ( O F ) . W_\Lambda(O)=\mu_\Lambda^F(O^F). W Λ ( O ) = μ Λ F ( O F ) . Proof. Each segment of F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ ) x 0 , x 1 , … , x k x_0,x_1,\ldots,x_k x 0 , x 1 , … , x k F F F g x 0 = 1 g_{x_0}=1 g x 0 = 1 g x i + 1 = g x i Q ( x i , x i + 1 ) g_{x_{i+1}}=g_{x_i}Q_{(x_i,x_{i+1})} g x i + 1 = g x i Q ( x i , x i + 1 ) Q e g = 1 Q_e^g=1 Q e g = 1 F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ ) F F F Q A ↦ Q A g Q_A\mapsto Q_A^g Q A ↦ Q A g O O O
No leftover holonomy appears because F ∩ E ( Λ ) F\cap E(\Lambda) F ∩ E ( Λ ) μ Λ F \mu_\Lambda^F μ Λ F γ F \gamma^F γ F G A G^A G A e ∈ A e\in A e ∈ A γ e F \gamma_e^F γ e F Q e Q_e Q e exp ( N β ∑ p ∋ e Re Tr Q p F ) \exp(N\beta\sum_{p\ni e}\operatorname{Re}\operatorname{Tr}Q_p^F) exp ( N β ∑ p ∋ e Re Tr Q p F ) γ F \gamma^F γ F δ \delta δ G G G D D D S U ( 2 ) SU(2) S U ( 2 ) S 3 S^3 S 3 D = 2 D=2 D = 2 S U ( N ) SU(N) S U ( N ) e ′ e' e ′ e e e
C e e ′ = sup W 1 ( γ e F ( ⋅ ∣ Q ) , γ e F ( ⋅ ∣ Q ′ ) ) δ ( Q e ′ , Q e ′ ′ ) , C_{ee'}=\sup\frac{W_1\left(\gamma_e^F(\mathord{\cdot}\mid Q),\gamma_e^F(\mathord{\cdot}\mid Q')\right)}{\delta(Q_{e'},Q'_{e'})}, C e e ′ = sup δ ( Q e ′ , Q e ′ ′ ) W 1 ( γ e F ( ⋅ ∣ Q ) , γ e F ( ⋅ ∣ Q ′ ) ) , the supremum over pairs of configurations that differ only at e ′ e' e ′ e e e e ′ e' e ′
Proposition 11. Suppose that every row and every column of ( C e e ′ ) (C_{ee'}) ( C e e ′ ) α < 1 \alpha<1 α < 1
(i) γ F \gamma^F γ F μ F \mu_F μ F
(ii) for every gauge-invariant local O O O ω ( O ) = lim W Λ ( O ) \omega(O)=\lim W_\Lambda(O) ω ( O ) = lim W Λ ( O ) Z 4 \mathbb{Z}^4 Z 4 μ F ( O F ) \mu_F(O^F) μ F ( O F )
(iii) ω \omega ω
(iv) if β ≥ 0 \beta\ge0 β ≥ 0 ω \omega ω
(v) for gauge-invariant local O , O ′ O,O' O , O ′ O O O ∣ Cov ω ( O , O ′ ) ∣ ≤ C O , O ′ e − λ dist ( O , O ′ ) \left|\operatorname{Cov}_\omega(O,O')\right|\le C_{O,O'}e^{-\lambda\operatorname{dist}(O,O')} ∣ Cov ω ( O , O ′ ) ∣ ≤ C O , O ′ e − λ dist ( O , O ′ ) λ < − log α \lambda<-\log\alpha λ < − log α
(vi) if β ≥ 0 \beta\ge0 β ≥ 0 ω \omega ω − log α -\log\alpha − log α
Figure 4. (a) The line field in the slice x 3 = 0 x_3=0 x 3 = 0 F F F Lemma 9 (i)), and the segments that leave the box are its boundary stubs. (b) One plaquette with its fixed link e 1 e_1 e 1 21 10 \frac{21}{10} 10 21 e 1 e_1 e 1 5 4 \frac{5}{4} 4 5 (7) once Y 1 = 0 Y_1=0 Y 1 = 0
Proof. (i) Gibbs measures exist by compactness, and the comparison lemma of Appendix B with S = ∅ S=\varnothing S = ∅
(ii) Extend μ Λ F \mu_\Lambda^F μ Λ F Λ \Lambda Λ S S S P ( Λ ) P(\Lambda) P ( Λ ) E ( Λ ) E(\Lambda) E ( Λ ) e ∉ S e\notin S e ∈ / S e e e P ( Λ ) P(\Lambda) P ( Λ ) Q e Q_e Q e γ e F \gamma_e^F γ e F T T T O F O^F O F λ < − log α \lambda<-\log\alpha λ < − log α e − λ dist ( ⋅ , T ) e^{-\lambda\operatorname{dist}(\cdot,T)} e − λ dist ( ⋅ , T )
∣ μ Λ F ( O F ) − μ F ( O F ) ∣ ≤ L θ D 1 − θ ∑ z ∈ S e − λ dist ( z , T ) , \left|\mu_\Lambda^F(O^F)-\mu_F(O^F)\right|\le L\frac{\theta D}{1-\theta}\sum_{z\in S}e^{-\lambda\operatorname{dist}(z,T)}, μ Λ F ( O F ) − μ F ( O F ) ≤ L 1 − θ θ D z ∈ S ∑ e − λ dist ( z , T ) , with θ = α e λ < 1 \theta=\alpha e^\lambda<1 θ = α e λ < 1 L L L O F O^F O F C T R 2 e − λ R C_TR^2e^{-\lambda R} C T R 2 e − λ R R R R T T T Λ \Lambda Λ O O O Lemma 10 gives W Λ ( O ) = μ Λ F ( O F ) W_\Lambda(O)=\mu_\Lambda^F(O^F) W Λ ( O ) = μ Λ F ( O F )
(iii) W Λ ( τ v O ) = W Λ − v ( O ) W_\Lambda(\tau_vO)=W_{\Lambda-v}(O) W Λ ( τ v O ) = W Λ − v ( O ) Λ − v \Lambda-v Λ − v Z 4 \mathbb{Z}^4 Z 4 W Λ W_\Lambda W Λ
(iv) For β ≥ 0 \beta\ge0 β ≥ 0 W Λ W_\Lambda W Λ [64 ] , and reflection positivity passes to the limit along such boxes, which is ω \omega ω
(v) Cov ω ( O , O ′ ) = Cov μ F ( O F , O ′ F ) \operatorname{Cov}_\omega(O,O')=\operatorname{Cov}_{\mu_F}(O^F,O'^F) Cov ω ( O , O ′ ) = Cov μ F ( O F , O ′ F ) F F F
(vi) Let ϑ \vartheta ϑ τ t \tau_t τ t O O O
∣ ω ( ϑ ( O ‾ ) τ t O ) − ∣ ω ( O ) ∣ 2 ∣ ≤ C O e − λ t . \left|\omega(\vartheta(\overline{O})\tau_tO)-|\omega(O)|^2\right|\le C_Oe^{-\lambda t}. ω ( ϑ ( O ) τ t O ) − ∣ ω ( O ) ∣ 2 ≤ C O e − λ t . On the support of O O O L 2 ( ω ) L^2(\omega) L 2 ( ω ) E Λ E^\Lambda E Λ L 2 L^2 L 2 [ 0 , e − λ ] [0,e^{-\lambda}] [ 0 , e − λ ] λ < − log α \lambda<-\log\alpha λ < − log α
The proposition replaces the torus limits and the uniqueness of the ungauged Gibbs measure used in [14 Thm. F] by the free-boundary limit, which has all the properties the reconstruction of the transfer matrix needs.10
IV SU(2) up to one third
Proof of Theorem 1 . Identify SU(2) with the unit quaternions and use the chordal metric. For an active link e e e p ∋ e p\ni e p ∋ e s p s_p s p Re ( Q e s p ) = 1 2 Tr Q p \operatorname{Re}(Q_es_p)=\frac{1}{2}\operatorname{Tr}Q_p Re ( Q e s p ) = 2 1 Tr Q p F F F Re Tr Q p = 1 2 Tr Q p \operatorname{Re}\operatorname{Tr}Q_p=\frac{1}{2}\operatorname{Tr}Q_p Re Tr Q p = 2 1 Tr Q p R 4 \mathbb{R}^4 R 4 Q e Q_e Q e ν b ( d x ) ∝ e ⟨ x , b ⟩ d x \nu_b(dx)\propto e^{\langle x,b\rangle}dx ν b ( d x ) ∝ e ⟨ x , b ⟩ d x S 3 S^3 S 3 b = 4 β ∑ p ∋ e s p ‾ b=4\beta\sum_{p\ni e}\overline{s_p} b = 4 β ∑ p ∋ e s p e ′ e' e ′ δ \delta δ b b b 4 ∣ β ∣ δ 4|\beta|\delta 4∣ β ∣ δ Theorem 125 ), W 1 ( ν b , ν b ′ ) ≤ 1 4 ∣ b − b ′ ∣ W_1(\nu_b,\nu_{b'}) \le\frac{1}{4}|b-b'| W 1 ( ν b , ν b ′ ) ≤ 4 1 ∣ b − b ′ ∣ C e e ′ ≤ ∣ β ∣ C_{ee'} \le|\beta| C e e ′ ≤ ∣ β ∣ Lemma 9 (iv) every row and column has twelve nonzero entries, so α ≤ 12 ∣ β ∣ = 3 ∣ β W ∣ \alpha\le12|\beta| = 3|\beta_W| α ≤ 12∣ β ∣ = 3∣ β W ∣ ∣ β W ∣ < 1 3 |\beta_W| < \frac{1}{3} ∣ β W ∣ < 3 1
Proof of Theorem 2. For the line field, every staple of an active link contains two active links, so the six staples can be chosen with sum zero while the staple through a given partner e ′ e' e ′ b = 0 b=0 b = 0 ν b \nu_b ν b h h h 1 4 ∣ h ∣ \frac{1}{4}|h| 4 1 ∣ h ∣ Lemma 124 (i), so C e e ′ = ∣ β ∣ C_{ee'} = |\beta| C e e ′ = ∣ β ∣ 12 ∣ β ∣ = 3 ∣ β W ∣ 12|\beta| = 3|\beta_W| 12∣ β ∣ = 3∣ β W ∣
Now let F ′ F' F ′ C ′ C' C ′ w w w C ′ w ≤ α w C'w \le\alpha w C ′ w ≤ α w Λ n \Lambda_n Λ n n n n C n C_n C n E ( Λ n ) E(\Lambda_n) E ( Λ n ) C n w ≤ α w C_nw \le\alpha w C n w ≤ α w ρ ( C n ) ≤ α \rho(C_n) \le\alpha ρ ( C n ) ≤ α C n C_n C n ρ ( C n ) \rho(C_n) ρ ( C n ) 4 n 4 − O ( n 3 ) 4n^4-O(n^3) 4 n 4 − O ( n 3 ) n 4 n^4 n 4 a a a 3 n 4 − O ( n 3 ) 3n^4-O(n^3) 3 n 4 − O ( n 3 ) a p a_p a p p p p P = 6 n 4 − O ( n 3 ) P=6n^4-O(n^3) P = 6 n 4 − O ( n 3 ) ∑ p a p = 6 a − O ( n 3 ) \sum_p a_p=6a-O(n^3) ∑ p a p = 6 a − O ( n 3 ) ∣ β ∣ ∑ p a p ( a p − 1 ) |\beta|\sum_p a_p(a_p-1) ∣ β ∣ ∑ p a p ( a p − 1 )
∣ β ∣ a ∑ p a p ( a p − 1 ) ≥ ∣ β ∣ P a a ˉ ( a ˉ − 1 ) = 6 ∣ β ∣ ( a ˉ − 1 ) − o ( 1 ) , \frac{|\beta|}{a}\sum_p a_p(a_p-1) \ge\frac{|\beta|P}{a}\bar a(\bar a-1)=6|\beta|(\bar a-1)-o(1), a ∣ β ∣ p ∑ a p ( a p − 1 ) ≥ a ∣ β ∣ P a ˉ ( a ˉ − 1 ) = 6∣ β ∣ ( a ˉ − 1 ) − o ( 1 ) , with a ˉ = ∑ p a p / P = 6 a / P − o ( 1 ) ≥ 3 − o ( 1 ) \bar a=\sum_p a_p/P=6a/P-o(1)\ge3-o(1) a ˉ = ∑ p a p / P = 6 a / P − o ( 1 ) ≥ 3 − o ( 1 ) α ≥ 12 ∣ β ∣ − o ( 1 ) \alpha\ge12|\beta|-o(1) α ≥ 12∣ β ∣ − o ( 1 ) n n n α ≥ 12 ∣ β ∣ = 3 ∣ β W ∣ \alpha\ge12|\beta|=3|\beta_W| α ≥ 12∣ β ∣ = 3∣ β W ∣
Without gauge fixing the same count gives eighteen partners and the ceiling β W = 2 9 \beta_W=\frac{2}{9} β W = 9 2 [14 Prop. 13] ; the temporal gauge, a family of lines in one direction, leaves fourteen. The line field is optimal among forests: twelve is the mean number of partners forced by the number of free links, and it is attained at every link.
SU(2) past one third: plaquette blocks
Throughout this section G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) ∣ x − y ∣ |x-y| ∣ x − y ∣ x ˉ \bar{x} x ˉ ⟨ x , y ⟩ = Re ( x y ˉ ) \langle x,y\rangle=\operatorname{Re}(x\bar y) ⟨ x , y ⟩ = Re ( x y ˉ ) R 4 \mathbb{R}^4 R 4 d x dx d x S 3 S^3 S 3 Re ( x y z ) = Re ( y z x ) = ⟨ y , z x ˉ ⟩ \operatorname{Re}(xyz)=\operatorname{Re}(yzx)=\langle y,z\bar{x}\rangle Re ( x y z ) = Re ( y z x ) = ⟨ y , z x ˉ ⟩
Z 0 ( c ) = ∫ S 3 e ⟨ c , x ⟩ d x = ∑ j ≥ 0 ( ∣ c ∣ 2 / 4 ) j j ! ( j + 1 ) ! , c ∈ R 4 , Z_0(c)=\int_{S^3}e^{\langle c,x\rangle}\,dx=\sum_{j\ge0}\frac{(|c|^2/4)^j}{j!(j+1)!},\qquad c\in\mathbb{R}^4, Z 0 ( c ) = ∫ S 3 e ⟨ c , x ⟩ d x = j ≥ 0 ∑ j ! ( j + 1 )! ( ∣ c ∣ 2 /4 ) j , c ∈ R 4 , let ν c ( d x ) ∝ e ⟨ c , x ⟩ d x \nu_c(dx)\propto e^{\langle c,x\rangle}dx ν c ( d x ) ∝ e ⟨ c , x ⟩ d x g = log Z 0 g=\log Z_0 g = log Z 0 ∇ g ( c ) \nabla g(c) ∇ g ( c ) Hess g ( c ) \operatorname{Hess}g(c) Hess g ( c ) ν c \nu_c ν c
The blocks
Let B \mathcal{B} B x x x { 0 , 1 } \{0,1\} { 0 , 1 } x 0 ≡ x 1 x_0\equiv x_1 x 0 ≡ x 1 { 2 , 3 } \{2,3\} { 2 , 3 } x 2 ≡ x 3 x_2\equiv x_3 x 2 ≡ x 3 2 Z 4 2\mathbb{Z}^4 2 Z 4 Figure 5 draws them). A block plaquette is a plaquette of B \mathcal{B} B
Figure 5. Block plaquettes, shaded, in (a) a plane { 0 , 1 } \{0,1\} { 0 , 1 } { 2 , 3 } \{2,3\} { 2 , 3 } x x x x 0 = x 1 x_0=x_1 x 0 = x 1 x 2 = x 3 x_2=x_3 x 2 = x 3 F F F
Lemma 12. The set B \mathcal{B} B 2 Z 4 2\mathbb{Z}^4 2 Z 4
(i) the blocks partition the active links;
(ii) every plaquette p ∉ B p\notin\mathcal{B} p ∈ / B
(iii) every active link lies in exactly five plaquettes outside B \mathcal{B} B B \mathcal{B} B
(iv) if an active link j j j B B B B \mathcal{B} B B B B 2 2 2 j j j
Proof. Invariance under 2 Z 4 2\mathbb{Z}^4 2 Z 4
Every link lies in exactly one plaquette of B \mathcal{B} B ( y , μ ) (y,\mu) ( y , μ ) μ ∈ { 0 , 1 } \mu\in\{0,1\} μ ∈ { 0 , 1 } { 2 , 3 } \{2,3\} { 2 , 3 } { 0 , 1 } \{0,1\} { 0 , 1 } y y y y − e ν y-e_\nu y − e ν ν \nu ν { 0 , 1 } \{0,1\} { 0 , 1 } y y y y − e ν y-e_\nu y − e ν ν \nu ν y 0 − y 1 y_0-y_1 y 0 − y 1 x 0 ≡ x 1 x_0\equiv x_1 x 0 ≡ x 1 2 , 3 2,3 2 , 3 Z 4 \mathbb{Z}^4 Z 4 B \mathcal{B} B
(ii) Let p ∉ B p\notin\mathcal{B} p ∈ / B e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 q i q_i q i e i e_i e i q i ≠ p q_i\ne p q i = p q i ∩ p = { e i } q_i\cap p=\{e_i\} q i ∩ p = { e i } q i = q k q_i=q_k q i = q k i ≠ k i\ne k i = k e i , e k e_i,e_k e i , e k p p p q i q_i q i p p p
(iii) An active link e e e Lemma 9 ), and by (i) exactly one of them is in B \mathcal{B} B e e e e e e e e e f f f p ∉ B p\notin\mathcal{B} p ∈ / B e e e p , e , f p,e,f p , e , f p p p e e e
(iv) Let f ∈ B f\in B f ∈ B p ∉ B p\notin\mathcal{B} p ∈ / B j j j 2 − 1 / 2 2^{-1/2} 2 − 1/2 1 1 1 j j j 1 1 1 f f f f f f 1 1 1 2 2 2
Fix a block plaquette p p p e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 1 2 Tr Q p = Re ( y 1 y 2 y 3 ) \frac{1}{2}\operatorname{Tr}Q_p=\operatorname{Re}(y_1y_2y_3) 2 1 Tr Q p = Re ( y 1 y 2 y 3 ) y i y_i y i Q e i Q_{e_i} Q e i Q e i − 1 Q_{e_i}^{-1} Q e i − 1 e i e_i e i e i e_i e i 4 β ⟨ y i , s ˉ ⟩ 4\beta\langle y_i,\bar{s}\rangle 4 β ⟨ y i , s ˉ ⟩ s s s y i y_i y i
μ b ( d y ) ∝ exp ( ∑ i = 1 3 ⟨ y i , b i ⟩ + 4 β Re ( y 1 y 2 y 3 ) ) d y 1 d y 2 d y 3 , ( 2 ) (2) \mu_b(dy)\propto\exp\left(\sum_{i=1}^{3}\langle y_i,b_i\rangle+4\beta\operatorname{Re}(y_1y_2y_3)\right)\,dy_1\,dy_2\,dy_3,\qquad(2) \tag*{(2)} μ b ( d y ) ∝ exp ( i = 1 ∑ 3 ⟨ y i , b i ⟩ + 4 β Re ( y 1 y 2 y 3 ) ) d y 1 d y 2 d y 3 , ( 2 ) (2) with b i b_i b i 4 β s ˉ 4\beta\bar{s} 4 β s ˉ ∣ b i ∣ ≤ 20 ∣ β ∣ |b_i|\leq20|\beta| ∣ b i ∣ ≤ 20∣ β ∣ S 3 S^3 S 3 y y y Q Q Q ( S 3 ) 3 (S^3)^3 ( S 3 ) 3 d ( y , y ′ ) = ∑ i ∣ y i − y i ′ ∣ d(y,y')=\sum_i|y_i-y_i'| d ( y , y ′ ) = ∑ i ∣ y i − y i ′ ∣
The sensitivity of one block
Let λ ∗ \lambda_* λ ∗ τ \tau τ Hess g ( c ) \operatorname{Hess}g(c) Hess g ( c ) ∣ c ∣ ≤ 24 ∣ β ∣ |c|\leq24|\beta| ∣ c ∣ ≤ 24∣ β ∣ M M M D 3 g ( c ) D^3g(c) D 3 g ( c )
w = 8 β 2 ( τ − λ ∗ ) , s = 1 4 + ( e w − 1 ) e w τ , w=8\beta^2(\tau-\lambda_*),\qquad s=\frac{1}{4}+(e^w-1)e^w\sqrt{\tau}, w = 8 β 2 ( τ − λ ∗ ) , s = 4 1 + ( e w − 1 ) e w τ , C = 20 ∣ β ∣ ( 4 ∣ β ∣ τ s + 16 β 2 M ) , K = ( 1 + ∣ β ∣ ) s 1 − C . (3) C=20|\beta|(4|\beta|\tau s+16\beta^2M),\qquad K=\frac{(1+|\beta|)s}{1-C}. \tag*{(3)} C = 20∣ β ∣ ( 4∣ β ∣ τ s + 16 β 2 M ) , K = 1 − C ( 1 + ∣ β ∣ ) s . (3) assuming C < 1 C<1 C < 1
Lemma 13. Let ν \nu ν λ = e R ν / ν ( e R ) \lambda=e^R\nu/\nu(e^R) λ = e R ν / ν ( e R ) osc R ≤ ω \operatorname{osc}R\leq\omega osc R ≤ ω f , h f,h f , h
Cov λ ( f , h ) ≤ Cov ν ( f , h ) + ( e ω − 1 ) e ω sd ν ( f ) sd ν ( h ) . \operatorname{Cov}_{\lambda}(f,h)\leq\operatorname{Cov}_{\nu}(f,h)+(e^\omega-1)e^\omega\operatorname{sd}_{\nu}(f)\operatorname{sd}_{\nu}(h). Cov λ ( f , h ) ≤ Cov ν ( f , h ) + ( e ω − 1 ) e ω sd ν ( f ) sd ν ( h ) . Proof. We may assume 0 ≤ R ≤ ω 0\leq R\leq\omega 0 ≤ R ≤ ω ρ = e R / ν ( e R ) \rho=e^R/\nu(e^R) ρ = e R / ν ( e R ) [ e − ω , e ω ] [e^{-\omega},e^\omega] [ e − ω , e ω ] δ = ∥ ρ − 1 ∥ ∞ ≤ e ω − 1 \delta=\|\rho-1\|_\infty\leq e^\omega-1 δ = ∥ ρ − 1 ∥ ∞ ≤ e ω − 1 f ~ = f − ν f \tilde f=f-\nu f f ~ = f − ν f h ~ = h − ν h \tilde h=h-\nu h h ~ = h − ν h
Cov λ ( f , h ) − Cov ν ( f , h ) = ν ( ( ρ − 1 ) f ~ h ~ ) − ν ( ( ρ − 1 ) f ~ ) ν ( ( ρ − 1 ) h ~ ) , \operatorname{Cov}_{\lambda}(f,h)-\operatorname{Cov}_{\nu}(f,h)
=\nu((\rho-1)\tilde f\tilde h)-\nu((\rho-1)\tilde f)\nu((\rho-1)\tilde h), Cov λ ( f , h ) − Cov ν ( f , h ) = ν (( ρ − 1 ) f ~ h ~ ) − ν (( ρ − 1 ) f ~ ) ν (( ρ − 1 ) h ~ ) , and the right side is at most ( δ + δ 2 ) sd ν ( f ) sd ν ( h ) (\delta+\delta^2)\operatorname{sd}_{\nu}(f)\operatorname{sd}_{\nu}(h) ( δ + δ 2 ) sd ν ( f ) sd ν ( h )
Lemma 14. Let β ≥ 0 \beta\geq0 β ≥ 0 a , c 0 ∈ R 4 a,c_0\in\mathbb{R}^4 a , c 0 ∈ R 4 ∣ a ∣ , ∣ c 0 ∣ ≤ 20 β |a|,|c_0|\leq20\beta ∣ a ∣ , ∣ c 0 ∣ ≤ 20 β
κ ( d x ) ∝ exp ( ⟨ x , a ⟩ + g ( c 0 + 4 β x ˉ ) ) d x on S 3 . \kappa(dx)\propto\exp\bigl(\langle x,a\rangle+g(c_0+4\beta\bar{x})\bigr)\,dx\quad\text{on }S^3. κ ( d x ) ∝ exp ( ⟨ x , a ⟩ + g ( c 0 + 4 β x ˉ ) ) d x on S 3 . For every 1 1 1 f f f h ∈ R 4 h\in\mathbb{R}^4 h ∈ R 4
(i) Cov κ ( f , ⟨ x , h ⟩ ) ≤ s ∣ h ∣ \operatorname{Cov}_{\kappa}(f,\langle x,h\rangle)\leq s|h| Cov κ ( f , ⟨ x , h ⟩) ≤ s ∣ h ∣
(ii) if c 0 c_0 c 0 k k k ∣ d d t κ ( f ) ∣ ≤ ( 4 β τ s + 16 β 2 M ) ∣ k ∣ \left|\frac{d}{dt}\kappa(f)\right|\leq(4\beta\tau s+16\beta^2M)|k| d t d κ ( f ) ≤ ( 4 β τ s + 16 β 2 M ) ∣ k ∣
Proof. (i) Taylor’s formula at c 0 c_0 c 0 g ( c 0 + 4 β u ) = g ( c 0 ) + 4 β ⟨ ∇ g ( c 0 ) , u ⟩ + R ( u ) g(c_0+4\beta u)=g(c_0)+4\beta\langle\nabla g(c_0),u\rangle+R(u) g ( c 0 + 4 β u ) = g ( c 0 ) + 4 β ⟨ ∇ g ( c 0 ) , u ⟩ + R ( u ) R ( u ) = 8 β 2 Hess g ( ξ ) ( u , u ) R(u)=8\beta^2\operatorname{Hess}g(\xi)(u,u) R ( u ) = 8 β 2 Hess g ( ξ ) ( u , u ) ξ \xi ξ ∣ ξ ∣ ≤ 24 β |\xi|\leq24\beta ∣ ξ ∣ ≤ 24 β u = x ˉ u=\bar{x} u = x ˉ R ( x ˉ ) ∈ [ 8 β 2 λ ∗ , 8 β 2 τ ] R(\bar{x})\in[8\beta^2\lambda_*,8\beta^2\tau] R ( x ˉ ) ∈ [ 8 β 2 λ ∗ , 8 β 2 τ ] osc R ≤ w \operatorname{osc}R\leq w osc R ≤ w ⟨ ∇ g ( c 0 ) , x ˉ ⟩ = ⟨ x , ∇ g ( c 0 ) ⟩ \langle\nabla g(c_0),\bar{x}\rangle=\langle x,\nabla g(c_0)\rangle ⟨ ∇ g ( c 0 ) , x ˉ ⟩ = ⟨ x , ∇ g ( c 0 )⟩ κ \kappa κ e R ν / ν ( e R ) e^R\nu/\nu(e^R) e R ν / ν ( e R ) ν = ν c \nu=\nu_c ν = ν c c = a + 4 β ∇ g ( c 0 ) c=a+4\beta\nabla g(c_0) c = a + 4 β ∇ g ( c 0 ) Cov ν ( f , ⟨ x , h ⟩ ) = d d t ν c + t h ( f ) ≤ 1 4 ∣ h ∣ \operatorname{Cov}_{\nu}(f,\langle x,h\rangle)=\frac{d}{dt}\nu_{c+th}(f)\leq\frac14|h| Cov ν ( f , ⟨ x , h ⟩) = d t d ν c + t h ( f ) ≤ 4 1 ∣ h ∣ 1 1 1 sd ≤ 1 \operatorname{sd}\leq1 sd ≤ 1 sd ν ⟨ x , h ⟩ ≤ τ ∣ h ∣ \operatorname{sd}_{\nu}\langle x,h\rangle\leq\sqrt{\tau}|h| sd ν ⟨ x , h ⟩ ≤ τ ∣ h ∣ ∣ c ∣ ≤ ∣ a ∣ + 4 β ≤ 24 β |c|\leq|a|+4\beta\leq24\beta ∣ c ∣ ≤ ∣ a ∣ + 4 β ≤ 24 β
(ii) The derivative is Cov κ ( f , ⟨ ∇ g ( c 0 + 4 β x ˉ ) , k ⟩ ) \operatorname{Cov}_{\kappa}(f,\langle\nabla g(c_0+4\beta\bar{x}),k\rangle) Cov κ ( f , ⟨ ∇ g ( c 0 + 4 β x ˉ ) , k ⟩) ∇ g ( c 0 + 4 β u ) = ∇ g ( c 0 ) + 4 β Hess g ( c 0 ) u + N ( u ) \nabla g(c_0+4\beta u)=\nabla g(c_0)+4\beta\operatorname{Hess}g(c_0)u+N(u) ∇ g ( c 0 + 4 β u ) = ∇ g ( c 0 ) + 4 β Hess g ( c 0 ) u + N ( u ) ⟨ x , h ⟩ \langle x,h\rangle ⟨ x , h ⟩ h = 4 β Hess g ( c 0 ) k h=4\beta\operatorname{Hess}g(c_0)k h = 4 β Hess g ( c 0 ) k ∣ h ∣ ≤ 4 β τ ∣ k ∣ |h|\leq4\beta\tau|k| ∣ h ∣ ≤ 4 β τ ∣ k ∣ 4 β τ s ∣ k ∣ 4\beta\tau s|k| 4 β τ s ∣ k ∣ ∣ u ∣ , ∣ u ′ ∣ ≤ 1 |u|,|u'|\leq1 ∣ u ∣ , ∣ u ′ ∣ ≤ 1 ∥ Hess g ( c 0 + 4 β v ) − Hess g ( c 0 ) ∥ ≤ 4 β M ∣ v ∣ ≤ 4 β M \|\operatorname{Hess}g(c_0+4\beta v)-\operatorname{Hess}g(c_0)\|\leq4\beta M|v|\leq4\beta M ∥ Hess g ( c 0 + 4 β v ) − Hess g ( c 0 ) ∥ ≤ 4 β M ∣ v ∣ ≤ 4 β M ∣ N ( u ) − N ( u ′ ) ∣ ≤ 16 β 2 M ∣ u − u ′ ∣ |N(u)-N(u')|\leq16\beta^2M|u-u'| ∣ N ( u ) − N ( u ′ ) ∣ ≤ 16 β 2 M ∣ u − u ′ ∣ ⟨ N ( x ˉ ) , k ⟩ \langle N(\bar{x}),k\rangle ⟨ N ( x ˉ ) , k ⟩ 16 β 2 M ∣ k ∣ 16\beta^2M|k| 16 β 2 M ∣ k ∣ f f f 16 β 2 M ∣ k ∣ 16\beta^2M|k| 16 β 2 M ∣ k ∣
Lemma 15. Let C < 1 C<1 C < 1 b b b b ′ b' b ′ ∣ b i ∣ , ∣ b i ′ ∣ ≤ 20 ∣ β ∣ |b_i|,|b_i'|\leq20|\beta| ∣ b i ∣ , ∣ b i ′ ∣ ≤ 20∣ β ∣ b i b_i b i
W 1 ( μ b , μ b ′ ) ≤ K ∣ b i − b i ′ ∣ W_1(\mu_b,\mu_{b'})\leq K|b_i-b_i'| W 1 ( μ b , μ b ′ ) ≤ K ∣ b i − b i ′ ∣ in the metric d d d
Proof. The map y 1 ↦ − y 1 y_1\mapsto-y_1 y 1 ↦ − y 1 μ b \mu_b μ b β \beta β − β -\beta − β b 1 b_1 b 1 − b 1 -b_1 − b 1
β ≥ 0 \beta\ge0 β ≥ 0 Re ( y 1 y 2 y 3 ) \operatorname{Re}(y_1y_2y_3) Re ( y 1 y 2 y 3 ) i = 1 i=1 i = 1
Let f f f d d d ε = ∣ b 1 − b 1 ′ ∣ \varepsilon= |b_1-b'_1| ε = ∣ b 1 − b 1 ′ ∣
The middle link. Given y 1 , y 3 y_1,y_3 y 1 , y 3 y 2 y_2 y 2 ν c \nu_c ν c c = b 2 + 4 β y 3 y 1 ‾ c=b_2+4\beta\,\overline{y_3y_1} c = b 2 + 4 β y 3 y 1 b 1 b_1 b 1 y 1 y_1 y 1 η \eta η c c c 4 β η 4\beta\eta 4 β η ν c \nu_c ν c β η \beta\eta β η W 1 W_1 W 1 F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] F(y_1,y_3)=\mathbb{E}[f\mid y_1,y_3] F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] ( 1 + β ) (1+\beta) ( 1 + β ) μ b ( f ) = μ b 13 ( F ) \mu_b(f)=\mu_b^{13}(F) μ b ( f ) = μ b 13 ( F ) μ b 13 \mu_b^{13} μ b 13 ( y 1 , y 3 ) (y_1,y_3) ( y 1 , y 3 )
μ b 13 ∝ e ⟨ y 1 , b 1 ⟩ + ⟨ y 3 , b 3 ⟩ Z 0 ( b 2 + 4 β y 3 y 1 ‾ ) . \mu_b^{13}\propto e^{\langle y_1,b_1\rangle+\langle y_3,b_3\rangle}Z_0(b_2+4\beta\,\overline{y_3y_1}). μ b 13 ∝ e ⟨ y 1 , b 1 ⟩ + ⟨ y 3 , b 3 ⟩ Z 0 ( b 2 + 4 β y 3 y 1 ) . Since ∣ b 2 + 4 β y 3 y 1 ‾ ∣ = ∣ b 2 y 3 + 4 β y 1 ‾ ∣ = ∣ y 1 b 2 + 4 β y 3 ‾ ∣ |b_2+4\beta\overline{y_3y_1}|=|b_2y_3+4\beta\overline{y_1}|=|y_1b_2+4\beta\overline{y_3}| ∣ b 2 + 4 β y 3 y 1 ∣ = ∣ b 2 y 3 + 4 β y 1 ∣ = ∣ y 1 b 2 + 4 β y 3 ∣
κ 1 ( d y 1 ∣ y 3 ) ∝ e ⟨ y 1 , b 1 ⟩ + g ( b 2 y 3 + 4 β y 1 ‾ ) d y 1 , \kappa_1(dy_1\mid y_3)\propto e^{\langle y_1,b_1\rangle+g(b_2y_3+4\beta\overline{y_1})}\,dy_1, κ 1 ( d y 1 ∣ y 3 ) ∝ e ⟨ y 1 , b 1 ⟩ + g ( b 2 y 3 + 4 β y 1 ) d y 1 , κ 3 ( d y 3 ∣ y 1 ) ∝ e ⟨ y 3 , b 3 ⟩ + g ( y 1 b 2 + 4 β y 3 ‾ ) d y 3 , \kappa_3(dy_3\mid y_1)\propto e^{\langle y_3,b_3\rangle+g(y_1b_2+4\beta\overline{y_3})}\,dy_3, κ 3 ( d y 3 ∣ y 1 ) ∝ e ⟨ y 3 , b 3 ⟩ + g ( y 1 b 2 + 4 β y 3 ) d y 3 , both of the form of Lemma 14 , with ∣ c 0 ∣ = ∣ b 2 ∣ |c_0|=|b_2| ∣ c 0 ∣ = ∣ b 2 ∣
Two sites. Integrating Lemma 14 (i) along the segment from b 1 b_1 b 1 b 1 ′ b'_1 b 1 ′ W 1 ( κ 1 , κ 1 ′ ) ≤ s ε W_1(\kappa_1,\kappa'_1)\le s\varepsilon W 1 ( κ 1 , κ 1 ′ ) ≤ s ε y 3 y_3 y 3 y 3 y_3 y 3 y 3 ′ y'_3 y 3 ′ c 0 = b 2 y 3 c_0=b_2y_3 c 0 = b 2 y 3 ∣ b 2 ∣ ∣ y 3 − y 3 ′ ∣ ≤ 20 β ∣ y 3 − y 3 ′ ∣ |b_2||y_3-y'_3|\le20\beta|y_3-y'_3| ∣ b 2 ∣∣ y 3 − y 3 ′ ∣ ≤ 20 β ∣ y 3 − y 3 ′ ∣ ∣ c 0 ∣ ≤ 20 β |c_0|\le20\beta ∣ c 0 ∣ ≤ 20 β Lemma 14 (ii) gives W 1 ( κ 1 ( ⋅ ∣ y 3 ) , κ 1 ( ⋅ ∣ y 3 ′ ) ) ≤ C ∣ y 3 − y 3 ′ ∣ W_1(\kappa_1(\cdot\mid y_3),\kappa_1(\cdot\mid y'_3))\le C|y_3-y'_3| W 1 ( κ 1 ( ⋅ ∣ y 3 ) , κ 1 ( ⋅ ∣ y 3 ′ )) ≤ C ∣ y 3 − y 3 ′ ∣ κ 3 \kappa_3 κ 3 b 1 b_1 b 1 π \pi π μ b 13 \mu_b^{13} μ b 13 μ b ′ 13 \mu_{b'}^{13} μ b ′ 13 d 1 + d 3 d_1+d_3 d 1 + d 3 d k = E π ∣ y k − y k ′ ∣ d_k=\mathbb{E}_\pi|y_k-y'_k| d k = E π ∣ y k − y k ′ ∣ ( y 1 , y 1 ′ ) (y_1,y'_1) ( y 1 , y 1 ′ ) κ 1 ( ⋅ ∣ y 3 ) \kappa_1(\cdot\mid y_3) κ 1 ( ⋅ ∣ y 3 ) κ 1 ′ ( ⋅ ∣ y 3 ′ ) \kappa'_1(\cdot\mid y'_3) κ 1 ′ ( ⋅ ∣ y 3 ′ ) d 1 ≤ s ε + C d 3 d_1\le s\varepsilon+Cd_3 d 1 ≤ s ε + C d 3 d 3 ≤ C d 1 d_3\le Cd_1 d 3 ≤ C d 1 d 1 ≤ s ε / ( 1 − C 2 ) d_1\le s\varepsilon/(1-C^2) d 1 ≤ s ε / ( 1 − C 2 )
∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + β ) ( d 1 + d 3 ) ≤ ( 1 + β ) s 1 − C ε = K ε . |\mu_b(f)-\mu_{b'}(f)|\le(1+\beta)(d_1+d_3)\le\frac{(1+\beta)s}{1-C}\varepsilon=K\varepsilon. ∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + β ) ( d 1 + d 3 ) ≤ 1 − C ( 1 + β ) s ε = K ε . The constants
Lemma 16. Write g ( c ) = φ ( ∣ c ∣ 2 ) g(c)=\varphi(|c|^2) g ( c ) = φ ( ∣ c ∣ 2 ) r = ∣ c ∣ r=|c| r = ∣ c ∣ c ≠ 0 c\ne0 c = 0 Hess g ( c ) \operatorname{Hess}g(c) Hess g ( c ) 2 φ ′ ( r 2 ) 2\varphi'(r^2) 2 φ ′ ( r 2 ) 2 φ ′ ( r 2 ) + 4 r 2 φ ′ ′ ( r 2 ) 2\varphi'(r^2)+4r^2\varphi''(r^2) 2 φ ′ ( r 2 ) + 4 r 2 φ ′′ ( r 2 ) D 3 g ( c ) D^3g(c) D 3 g ( c )
max ( ∣ A + Γ ∣ , 2 3 ∣ A ∣ ) , A = 12 r φ ′ ′ ( r 2 ) , Γ = 8 r 3 φ ′ ′ ′ ( r 2 ) . \max\left(|A+\Gamma|,\frac{2}{3}|A|\right),\qquad A=12r\varphi''(r^2),\qquad\Gamma=8r^3\varphi'''(r^2). max ( ∣ A + Γ∣ , 3 2 ∣ A ∣ ) , A = 12 r φ ′′ ( r 2 ) , Γ = 8 r 3 φ ′′′ ( r 2 ) . On the ball r ≤ 2.07 r\le2.07 r ≤ 2.07 Hess g \operatorname{Hess}g Hess g 0.158491 0.158491 0.158491 0.25000001 0.25000001 0.25000001 D 3 g D^3g D 3 g 0.0615586 0.0615586 0.0615586
Proof. Differentiating φ ( ∣ c ∣ 2 ) \varphi(|c|^2) φ ( ∣ c ∣ 2 ) D 2 g ( v , v ) = 2 φ ′ ∣ v ∣ 2 + 4 φ ′ ′ ⟨ c , v ⟩ 2 D^2g(v,v)=2\varphi'|v|^2+4\varphi''\langle c,v\rangle^2 D 2 g ( v , v ) = 2 φ ′ ∣ v ∣ 2 + 4 φ ′′ ⟨ c , v ⟩ 2 D 3 g ( v , v , v ) = 12 φ ′ ′ ⟨ c , v ⟩ ∣ v ∣ 2 + 8 φ ′ ′ ′ ⟨ c , v ⟩ 3 D^3g(v,v,v)=12\varphi''\langle c,v\rangle|v|^2+8\varphi'''\langle c,v\rangle^3 D 3 g ( v , v , v ) = 12 φ ′′ ⟨ c , v ⟩ ∣ v ∣ 2 + 8 φ ′′′ ⟨ c , v ⟩ 3 ∣ v ∣ = 1 |v|=1 ∣ v ∣ = 1 t = ⟨ c / r , v ⟩ ∈ [ − 1 , 1 ] t=\langle c/r,v\rangle\in[-1,1] t = ⟨ c / r , v ⟩ ∈ [ − 1 , 1 ] A t + Γ t 3 At+\Gamma t^3 A t + Γ t 3 t = ± 1 t=\pm1 t = ± 1 2 3 ∣ A ∣ ∣ t ∣ ≤ 2 3 ∣ A ∣ \frac{2}{3}|A||t|\le\frac{2}{3}|A| 3 2 ∣ A ∣∣ t ∣ ≤ 3 2 ∣ A ∣ [8 ] the supremum of a symmetric trilinear form on the diagonal of the unit sphere is its norm. The numerical bounds are obtained in ball arithmetic [48 ] from the series of Z 0 Z_0 Z 0 2 ⋅ 10 4 2\cdot10^4 2 ⋅ 1 0 4 [ 0 , 2.07 ] [0,2.07] [ 0 , 2.07 ]
The proof of Theorem 3
Let γ B B \gamma_B^{\mathcal{B}} γ B B B B B γ B B \gamma_B^{\mathcal{B}} γ B B B B B (2) . It comes from the same interaction as γ F \gamma^{\mathcal{F}} γ F
C B j = sup W 1 ( γ B B ( ⋅ ∣ Q ) , γ B B ( ⋅ ∣ Q ′ ) ) ∣ Q j − Q j ′ ∣ C_{Bj}=\sup\frac{W_1(\gamma_B^{\mathcal{B}}(\cdot\mid Q),\gamma_B^{\mathcal{B}}(\cdot\mid Q'))}{|Q_j-Q'_j|} C B j = sup ∣ Q j − Q j ′ ∣ W 1 ( γ B B ( ⋅ ∣ Q ) , γ B B ( ⋅ ∣ Q ′ )) for an active link j ∉ B j\notin B j ∈ / B j j j W 1 W_1 W 1 d d d
Proposition 17. Suppose that ∑ B C B j ≤ α < 1 \sum_B C_{Bj}\le\alpha<1 ∑ B C B j ≤ α < 1 j j j − 1 2 log α -\frac{1}{2}\log\alpha − 2 1 log α − log α -\log\alpha − log α
Proof. The proof of Proposition 11 goes through with Lemma 114 of Appendix B in place of Lemma 112, and with the set S S S H H H
Proof of Theorem 3 . Let ∣ β ∣ ≤ 0.345 / 4 |\beta|\le0.345/4 ∣ β ∣ ≤ 0.345/4 r ≤ 2.07 r\le2.07 r ≤ 2.07 ∣ c ∣ ≤ 24 ∣ β ∣ |c|\le24|\beta| ∣ c ∣ ≤ 24∣ β ∣ (3) is increasing in ∣ β ∣ |\beta| ∣ β ∣
α ( β W ) = 40 ∣ β ∣ K . (4) \alpha(\beta_W)=40|\beta|K. \tag*{(4)} α ( β W ) = 40∣ β ∣ K . (4) Changing an active link j j j η \eta η b j b_j b j 4 ∣ β ∣ ∣ η ∣ 4|\beta||\eta| 4∣ β ∣∣ η ∣ Lemma 15 , applied once for each of these ten fields, gives ∑ B C B j ≤ 40 ∣ β ∣ K \sum_B C_{Bj}\le40|\beta|K ∑ B C B j ≤ 40∣ β ∣ K (3) in ball arithmetic at β W = 0.345 \beta_W=0.345 β W = 0.345 w ≤ 0.0054459 w\le0.0054459 w ≤ 0.0054459 s ≤ 0.252746 s\le0.252746 s ≤ 0.252746 C ≤ 0.050243 C\le0.050243 C ≤ 0.050243 K ≤ 0.289069 K\le0.289069 K ≤ 0.289069 α ≤ 0.99729 \alpha\le0.99729 α ≤ 0.99729 β W = 1 3 \beta_W=\frac13 β W = 3 1 α ≤ 0.95602 \alpha\le0.95602 α ≤ 0.95602
Remark 18. Most of the range is kept. A second bound for the same sensitivity, which keeps the law of the middle link instead of linearising it, evaluated in floating point on a quadrature grid and maximised over the fields b b b 0.2783 0.2783 0.2783 β W = 0.34 \beta_W=0.34 β W = 0.34 0.2881 0.2881 0.2881 Lemma 15 , and 40 ∣ β ∣ 40|\beta| 40∣ β ∣ β W = 0.357 \beta_W=0.357 β W = 0.357
Curvature in the gauge
In this section G ⊂ U ( N ) G\subset U(N) G ⊂ U ( N ) g \mathfrak{g} g ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y ) Theorem 4 is proved first for U ( N ) U(N) U ( N ) Q Q Q v = ( X e Q e ) v=(X_eQ_e) v = ( X e Q e ) ∣ v ∣ 2 = ∑ e ∣ X e ∣ 2 |v|^2=\sum_e|X_e|^2 ∣ v ∣ 2 = ∑ e ∣ X e ∣ 2 x x x μ < ν \mu<\nu μ < ν e 1 = ( x , μ ) e_1=(x,\mu) e 1 = ( x , μ ) e 2 = ( x + e μ , ν ) e_2=(x+e_\mu,\nu) e 2 = ( x + e μ , ν ) e 3 = ( x + e ν , μ ) e_3=(x+e_\nu,\mu) e 3 = ( x + e ν , μ ) e 4 = ( x , ν ) e_4=(x,\nu) e 4 = ( x , ν ) x x x
Y 1 = X e 1 , Y 2 = Q e 1 X e 2 Q e 1 ∗ , Y 3 = Q e 4 X e 3 Q e 4 ∗ , Y 4 = X e 4 , (5) Y_1=X e_1,\quad Y_2=Q_{e_1}X_{e_2}Q_{e_1}^{*},\quad Y_3=Q_{e_4}X_{e_3}Q_{e_4}^{*},\quad Y_4=Xe_4, \tag*{(5)} Y 1 = X e 1 , Y 2 = Q e 1 X e 2 Q e 1 ∗ , Y 3 = Q e 4 X e 3 Q e 4 ∗ , Y 4 = X e 4 , (5) and write W = Q p W=Q_p W = Q p Re Tr Q p \operatorname{Re}\operatorname{Tr}Q_p Re Tr Q p Q e ( t ) = e t X e Q e Q_e(t)=e^{tX_e}Q_e Q e ( t ) = e t X e Q e
f p = ∑ i = 1 4 Re Tr ( Y i 2 W ) + 2 Re Tr ( Y 1 Y 2 W ) + 2 Re Tr ( Y 3 Y 4 W ) − 2 ∑ i ∈ { 1 , 2 } , j ∈ { 3 , 4 } Re Tr ( Y i W Y j ) . (6) f_p=\sum_{i=1}^{4}\operatorname{Re}\operatorname{Tr}(Y_i^2W)+2\operatorname{Re}\operatorname{Tr}(Y_1Y_2W)+2\operatorname{Re}\operatorname{Tr}(Y_3Y_4W)-2\sum_{\substack{i\in\{1,2\},\,j\in\{3,4\}}}\operatorname{Re}\operatorname{Tr}(Y_iWY_j). \tag*{(6)} f p = i = 1 ∑ 4 Re Tr ( Y i 2 W ) + 2 Re Tr ( Y 1 Y 2 W ) + 2 Re Tr ( Y 3 Y 4 W ) − 2 i ∈ { 1 , 2 } , j ∈ { 3 , 4 } ∑ Re Tr ( Y i W Y j ) . (6) so that Hess S 0 ( v , v ) = ∑ p f p \operatorname{Hess}S_0(v,v)=\sum_p f_p Hess S 0 ( v , v ) = ∑ p f p Q p = Q e 1 Q e 2 Q e 3 ∗ Q e 4 ∗ Q_p=Q_{e_1}Q_{e_2}Q_{e_3}^{*}Q_{e_4}^{*} Q p = Q e 1 Q e 2 Q e 3 ∗ Q e 4 ∗ Q e 1 e t X e 2 = e t Y 2 Q e 1 Q_{e_1}e^{tX_{e_2}}=e^{tY_2}Q_{e_1} Q e 1 e t X e 2 = e t Y 2 Q e 1 e − t X e 3 Q e 4 ∗ = Q e 4 ∗ e − t Y 3 e^{-tX_{e_3}}Q_{e_4}^{*}=Q_{e_4}^{*}e^{-tY_3} e − t X e 3 Q e 4 ∗ = Q e 4 ∗ e − t Y 3 Q p ( t ) = e t Y 1 e t Y 2 W e − t Y 3 e − t Y 4 Q_p(t)=e^{tY_1}e^{tY_2}We^{-tY_3}e^{-tY_4} Q p ( t ) = e t Y 1 e t Y 2 W e − t Y 3 e − t Y 4 t = 0 t=0 t = 0 W W W W Y 3 Y 4 WY_3Y_4 W Y 3 Y 4 p p p
c p = 2 ⟨ Y 1 , Y 4 ⟩ − 2 ⟨ Y 1 , Y 2 ⟩ + 2 ⟨ Y 2 , W Y 3 W ∗ ⟩ − 2 ⟨ Y 3 , Y 4 ⟩ . (7) c_p=2\langle Y_1,Y_4\rangle-2\langle Y_1,Y_2\rangle+2\langle Y_2,WY_3W^{*}\rangle-2\langle Y_3,Y_4\rangle. \tag*{(7)} c p = 2 ⟨ Y 1 , Y 4 ⟩ − 2 ⟨ Y 1 , Y 2 ⟩ + 2 ⟨ Y 2 , W Y 3 W ∗ ⟩ − 2 ⟨ Y 3 , Y 4 ⟩ . (7) It collects the inner products of the transported vectors of two links that meet at a right angle at a corner of p p p
Lemma 19. On Z 4 \mathbb{Z}^4 Z 4 Q Q Q X X X
∑ y [ ∣ D y X ∣ 2 + ∑ μ ∣ X y , μ + Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ∣ 2 ] = 4 ∣ X ∣ 2 + ∑ p c p . \sum_y\left[|D_yX|^2+\sum_\mu\left|X_{y,\mu}+Q_{y-e_\mu,\mu}^{*}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}\right|^2\right]=4|X|^2+\sum_p c_p. y ∑ [ ∣ D y X ∣ 2 + μ ∑ X y , μ + Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ 2 ] = 4∣ X ∣ 2 + p ∑ c p . where D y X = ∑ μ [ X y , μ − Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ] D_yX=\sum_\mu[X_{y,\mu}-Q_{y-e_\mu,\mu}^{*}X_{y-e_\mu,\mu}Q_{y-e_\mu,\mu}] D y X = ∑ μ [ X y , μ − Q y − e μ , μ ∗ X y − e μ , μ Q y − e μ , μ ]
Proof. At a site y y y o μ o_\mu o μ μ \mu μ i μ i_\mu i μ D y X = ∑ μ ( o μ − i μ ) D_yX=\sum_\mu(o_\mu-i_\mu) D y X = ∑ μ ( o μ − i μ ) ∣ o μ ∣ 2 + ∣ i μ ∣ 2 |o_\mu|^2+|i_\mu|^2 ∣ o μ ∣ 2 + ∣ i μ ∣ 2 ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 ∣ o μ + i μ ∣ 2 |o_\mu+i_\mu|^2 ∣ o μ + i μ ∣ 2 4 ∣ X ∣ 2 4|X|^2 4∣ X ∣ 2 − 2 ⟨ o μ , i μ ⟩ -2\langle o_\mu,i_\mu\rangle − 2 ⟨ o μ , i μ ⟩ ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 + 2 ⟨ o μ , i μ ⟩ +2\langle o_\mu,i_\mu\rangle + 2 ⟨ o μ , i μ ⟩ ∣ o μ + i μ ∣ 2 |o_\mu+i_\mu|^2 ∣ o μ + i μ ∣ 2 ∣ D y X ∣ 2 |D_yX|^2 ∣ D y X ∣ 2 y y y (7) (this is the proof of [14 Lemma 1] ). In a box two perpendicular ends at y y y
Lemma 20. Let k ∈ { 1 , 2 , 3 , 4 } k\in\{1,2,3,4\} k ∈ { 1 , 2 , 3 , 4 } k ± 1 ( m o d 4 ) k\pm1\pmod4 k ± 1 ( mod 4 ) k k k k + 2 k+2 k + 2 N N N W ∈ U ( N ) W\in U(N) W ∈ U ( N ) Y 1 , … , Y 4 ∈ u ( N ) Y_1,\ldots,Y_4\in\mathfrak{u}(N) Y 1 , … , Y 4 ∈ u ( N ) Y k = 0 Y_k=0 Y k = 0
± f p + 9 10 c p ≤ 21 10 ( ∣ Y k − 1 ∣ 2 + ∣ Y k + 1 ∣ 2 ) + 5 4 ∣ Y k + 2 ∣ 2 . \pm f_p+\frac{9}{10}c_p\le\frac{21}{10}\left(|Y_{k-1}|^2+|Y_{k+1}|^2\right)+\frac{5}{4}|Y_{k+2}|^2. ± f p + 10 9 c p ≤ 10 21 ( ∣ Y k − 1 ∣ 2 + ∣ Y k + 1 ∣ 2 ) + 4 5 ∣ Y k + 2 ∣ 2 . Proof. Replacing W W W − W -W − W f p f_p f p c p c_p c p + + + W = diag ( w a ) W=\operatorname{diag}(w_a) W = diag ( w a ) F = RHS − f p − 9 10 c p \mathcal{F}=\operatorname{RHS}-f_p-\frac{9}{10}c_p F = RHS − f p − 10 9 c p a a a y i = − i ( Y i ) a a y_i=-i(Y_i)_{aa} y i = − i ( Y i ) aa a ≠ b a\ne b a = b z i = ( Y i ) a b z_i=(Y_i)_{ab} z i = ( Y i ) ab ( Y i ) b a = − z i ‾ (Y_i)_{ba}=-\overline{z_i} ( Y i ) ba = − z i
Proof of the bound in Theorem 4 . Every plaquette has exactly one link of F F F X X X Lemma 9 (iii) each active link is adjacent to the fixed link in four of its plaquettes and opposite to it in two. Summing Lemma 20 over the plaquettes,
∑ p ( ± f p + 9 10 c p ) ≤ ( 4 ⋅ 21 10 + 2 ⋅ 5 4 ) ∣ X ∣ 2 = 109 10 ∣ X ∣ 2 . \sum_p\left(\pm f_p+\frac{9}{10}c_p\right)\le\left(4\cdot\frac{21}{10}+2\cdot\frac{5}{4}\right)|X|^2=\frac{109}{10}|X|^2. p ∑ ( ± f p + 10 9 c p ) ≤ ( 4 ⋅ 10 21 + 2 ⋅ 4 5 ) ∣ X ∣ 2 = 10 109 ∣ X ∣ 2 . By Lemma 19, 9 10 ∑ p c p = 9 10 Σ − 36 10 ∣ X ∣ 2 \frac{9}{10}\sum_p c_p=\frac{9}{10}\Sigma-\frac{36}{10}|X|^2 10 9 ∑ p c p = 10 9 Σ − 10 36 ∣ X ∣ 2 Σ ≥ 0 \Sigma\ge0 Σ ≥ 0
± Hess S 0 ( v , v ) + 9 10 Σ ≤ ( 109 10 + 36 10 ) ∣ X ∣ 2 = 29 2 ∣ X ∣ 2 . \pm\operatorname{Hess}S_0(v,v)+\frac{9}{10}\Sigma\le\left(\frac{109}{10}+\frac{36}{10}\right)|X|^2=\frac{29}{2}|X|^2. ± Hess S 0 ( v , v ) + 10 9 Σ ≤ ( 10 109 + 10 36 ) ∣ X ∣ 2 = 2 29 ∣ X ∣ 2 . In a box, links near the boundary lie in fewer plaquettes and receive less. □
The factor 9 10 \frac{9}{10} 10 9 36 10 \frac{36}{10} 10 36 [14 Lemma 1] with the collinear term written as a square, so nothing is lost by Cauchy–Schwarz.
The two lemmas concern vectors in u ( N ) \mathfrak{u}(N) u ( N ) U ( N ) U(N) U ( N ) G ⊂ U ( N ) G\subset U(N) G ⊂ U ( N ) X e X_e X e g \mathfrak{g} g S 0 S_0 S 0 G E G^E G E G G G Q e Q_e Q e e t X Q e e^{tX}Q_e e tX Q e X ∈ g X\in\mathfrak{g} X ∈ g U ( N ) U(N) U ( N )
Lemma 21. Let G ⊂ U ( N ) G\subset U(N) G ⊂ U ( N ) g \mathfrak{g} g t ⊂ g \mathfrak{t}\subset\mathfrak{g} t ⊂ g ad H \operatorname{ad}H ad H g ⊗ C \mathfrak{g}\otimes\mathbb{C} g ⊗ C H H H C N \mathbb{C}^N C N H ∈ t H\in\mathfrak{t} H ∈ t i α ( H ) i\alpha(H) i α ( H ) i μ ( H ) i\mu(H) i μ ( H ) α \alpha α μ \mu μ C N \mathbb{C}^N C N Ric G = κ G ⟨ ⋅ , ⋅ ⟩ \operatorname{Ric}_G=\kappa_G\langle\cdot,\cdot\rangle Ric G = κ G ⟨ ⋅ , ⋅ ⟩
κ G = ∑ α α ( H ) 2 4 ∑ μ μ ( H ) 2 ( 0 ≠ H ∈ t ) . \kappa_G=\frac{\sum_\alpha\alpha(H)^2}{4\sum_\mu\mu(H)^2}\qquad(0\ne H\in\mathfrak{t}). κ G = 4 ∑ μ μ ( H ) 2 ∑ α α ( H ) 2 ( 0 = H ∈ t ) . Proof. For a bi-invariant metric the curvature is R ( X , Y ) Z = − 1 4 [ [ X , Y ] , Z ] R(X,Y)Z=-\frac14[[X,Y],Z] R ( X , Y ) Z = − 4 1 [[ X , Y ] , Z ] [60 Sect. 7] , so for an orthonormal basis ( E i ) (E_i) ( E i ) g \mathfrak{g} g
Ric ( X , X ) = ∑ i ⟨ R ( E i , X ) X , E i ⟩ = 1 4 ∑ i ∣ [ X , E i ] ∣ 2 = − 1 4 Tr ( ad X ) 2 , \operatorname{Ric}(X,X)=\sum_i\langle R(E_i,X)X,E_i\rangle
=\frac14\sum_i|[X,E_i]|^2=-\frac14\operatorname{Tr}(\operatorname{ad}X)^2, Ric ( X , X ) = i ∑ ⟨ R ( E i , X ) X , E i ⟩ = 4 1 i ∑ ∣ [ X , E i ] ∣ 2 = − 4 1 Tr ( ad X ) 2 , since ad X \operatorname{ad}X ad X B ( X , Y ) = Tr ( ad X ad Y ) B(X,Y)=\operatorname{Tr}(\operatorname{ad}X\operatorname{ad}Y) B ( X , Y ) = Tr ( ad X ad Y ) ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ g \mathfrak{g} g B = − 4 κ G ⟨ ⋅ , ⋅ ⟩ B=-4\kappa_G\langle\cdot,\cdot\rangle B = − 4 κ G ⟨ ⋅ , ⋅ ⟩ κ G \kappa_G κ G Ric = κ G ⟨ ⋅ , ⋅ ⟩ \operatorname{Ric}=\kappa_G\langle\cdot,\cdot\rangle Ric = κ G ⟨ ⋅ , ⋅ ⟩ H ∈ t H\in\mathfrak{t} H ∈ t B ( H , H ) = − ∑ α α ( H ) 2 B(H,H)=-\sum_\alpha\alpha(H)^2 B ( H , H ) = − ∑ α α ( H ) 2 ⟨ H , H ⟩ = − Tr H 2 = ∑ μ μ ( H ) 2 \langle H,H\rangle=-\operatorname{Tr}H^2=\sum_\mu\mu(H)^2 ⟨ H , H ⟩ = − Tr H 2 = ∑ μ μ ( H ) 2
Table I evaluates the ratio of Lemma 21 for every compact simple Lie algebra and a faithful representation of least dimension, on the root systems of [46 Sect. 12.1] ; the weights of the representations of dimensions 27 and 56 of E 6 E_6 E 6 E 7 E_7 E 7 E 8 E_8 E 8 A 2 A_2 A 2 A 1 A_1 A 1 248 = ( 78 , 1 ) + ( 27 , 3 ) + ( 27 , 3 ‾ ) + ( 1 , 8 ) 248=(78,1)+(27,3)+(27,\overline{3})+(1,8) 248 = ( 78 , 1 ) + ( 27 , 3 ) + ( 27 , 3 ) + ( 1 , 8 ) 248 = ( 133 , 1 ) + ( 56 , 2 ) + ( 1 , 3 ) 248=(133,1)+(56,2)+(1,3) 248 = ( 133 , 1 ) + ( 56 , 2 ) + ( 1 , 3 ) [70 ] . Taking for H H H θ ∨ \theta^\vee θ ∨ ∑ α α ( θ ∨ ) 2 = 4 h ∨ \sum_\alpha\alpha(\theta^\vee)^2=4h^\vee ∑ α α ( θ ∨ ) 2 = 4 h ∨ ∑ μ μ ( θ ∨ ) 2 = 2 ℓ ρ \sum_\mu\mu(\theta^\vee)^2=2\ell_\rho ∑ μ μ ( θ ∨ ) 2 = 2 ℓ ρ κ G = h ∨ / ( 2 ℓ ρ ) \kappa_G=h^\vee/(2\ell_\rho) κ G = h ∨ / ( 2 ℓ ρ ) S U ( N ) \mathrm{SU}(N) SU ( N ) g \mathfrak{g} g ρ \rho ρ g \mathfrak{g} g
Lemma 22. Let e ≠ e ′ e\ne e' e = e ′ p p p
Hess Re Tr Q p ( v , v ) = ∑ e , e ′ ∈ p H e e ′ p ( X e , X e ′ ) \operatorname{Hess}\operatorname{Re}\operatorname{Tr}Q_p(v,v)=\sum_{e,e'\in p}H^p_{ee'}(X_e,X_{e'}) Hess Re Tr Q p ( v , v ) = e , e ′ ∈ p ∑ H e e ′ p ( X e , X e ′ ) with bilinear forms satisfying H e ′ e p ( a , b ) = H e e ′ p ( b , a ) H^p_{e'e}(a,b)=H^p_{ee'}(b,a) H e ′ e p ( a , b ) = H e e ′ p ( b , a ) ∣ H e e ′ p ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ |H^p_{ee'}(a,b)|\le|a|\,|b| ∣ H e e ′ p ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ a , b ∈ u ( N ) a,b\in\mathfrak{u}(N) a , b ∈ u ( N )
Proof. By (6) the terms of f p f_p f p i ≠ j i\ne j i = j 2 Re Tr ( Y 1 Y 2 W ) 2\operatorname{Re}\operatorname{Tr}(Y_1Y_2W) 2 Re Tr ( Y 1 Y 2 W ) 2 Re Tr ( Y 3 Y 4 W ) 2\operatorname{Re}\operatorname{Tr}(Y_3Y_4W) 2 Re Tr ( Y 3 Y 4 W ) − 2 Re Tr ( Y i W Y j ) -2\operatorname{Re}\operatorname{Tr}(Y_iWY_j) − 2 Re Tr ( Y i W Y j ) H e e ′ p H^p_{ee'} H e e ′ p ± Re Tr ( a b W ) \pm\operatorname{Re}\operatorname{Tr}(abW) ± Re Tr ( abW ) ± Re Tr ( a W b ) \pm\operatorname{Re}\operatorname{Tr}(aWb) ± Re Tr ( aW b ) a a a b b b ∣ Tr ( a b W ) ∣ ≤ ∥ a ∥ H S ∥ b W ∥ H S = ∣ a ∣ ∣ b ∣ |\operatorname{Tr}(abW)|\le\|a\|_{\mathrm{HS}}\|bW\|_{\mathrm{HS}}=|a|\,|b| ∣ Tr ( abW ) ∣ ≤ ∥ a ∥ HS ∥ bW ∥ HS = ∣ a ∣ ∣ b ∣
Corollary 23. Let G ⊂ U ( N ) G\subset\mathrm{U}(N) G ⊂ U ( N ) N ∣ β ∣ < 2 29 κ G N|\beta|<\frac{2}{29}\kappa_G N ∣ β ∣ < 29 2 κ G μ Λ F \mu_\Lambda^F μ Λ F K = κ G − 29 2 N ∣ β ∣ > 0 K=\kappa_G-\frac{29}{2}N|\beta|>0 K = κ G − 2 29 N ∣ β ∣ > 0 2 / K 2/K 2/ K 1 / K 1/K 1/ K
Proof. The set { Q F = 1 } \{Q_F=1\} { Q F = 1 } κ G \kappa_G κ G N β Hess S 0 N\beta\operatorname{Hess}S_0 N β Hess S 0 X F = 0 X_F=0 X F = 0 [4 Cor. 5.7.2] .
Decay with an explicit rate
Let A ′ A' A ′ P ′ P' P ′ A ′ A' A ′ S = ∑ p ∈ P ′ c p Re Tr Q p F S=\sum_{p\in P'}c_p\operatorname{Re}\operatorname{Tr}Q_p^F S = ∑ p ∈ P ′ c p Re Tr Q p F 0 ≤ c p ≤ 1 0\le c_p\le1 0 ≤ c p ≤ 1 ∣ Hess S ( v , v ) ∣ ≤ 29 2 ∣ v ∣ 2 |\operatorname{Hess}S(v,v)|\le\frac{29}{2}|v|^2 ∣ Hess S ( v , v ) ∣ ≤ 2 29 ∣ v ∣ 2 v v v G A ′ G^{A'} G A ′ Lemma 9 (iv) the mixed blocks of its Hessian satisfy ∣ Hess S e e ′ ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ |\operatorname{Hess}S_{ee'}(a,b)|\le|a|\,|b| ∣ Hess S e e ′ ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ S Λ = ∑ p ∈ P ( Λ ) Re Tr Q p F S_\Lambda=\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p^F S Λ = ∑ p ∈ P ( Λ ) Re Tr Q p F μ \mu μ G A ′ G^{A'} G A ′ e N β S e^{N\beta S} e N β S L = Δ + N β ⟨ ∇ S , ∇ ⋅ ⟩ L=\Delta+N\beta\langle\nabla S,\nabla\cdot\rangle L = Δ + N β ⟨ ∇ S , ∇ ⋅ ⟩ L 2 ( μ ) L^2(\mu) L 2 ( μ ) P t = e t L P_t=e^{tL} P t = e t L ∇ e \nabla_e ∇ e e e e w : A ′ → ( 0 , ∞ ) w:A'\to(0,\infty) w : A ′ → ( 0 , ∞ ) Γ w ( f ) = ∑ e w e ∣ ∇ e f ∣ 2 \Gamma^w(f)=\sum_e w_e|\nabla_e f|^2 Γ w ( f ) = ∑ e w e ∣ ∇ e f ∣ 2 Γ \Gamma Γ w ≡ 1 w\equiv1 w ≡ 1
Lemma 24. Let λ ≥ 0 \lambda\ge0 λ ≥ 0 w e = e λ d e w_e=e^{\lambda d_e} w e = e λ d e ∣ d e − d e ′ ∣ ≤ 1 |d_e-d_{e'}|\le1 ∣ d e − d e ′ ∣ ≤ 1 e e e e ′ e' e ′
K λ = κ G − N ∣ β ∣ ( 29 2 + 12 ( cosh λ 2 − 1 ) ) . K_\lambda=\kappa_G-N|\beta|\left(\frac{29}{2}+12\left(\cosh\frac{\lambda}{2}-1\right)\right). K λ = κ G − N ∣ β ∣ ( 2 29 + 12 ( cosh 2 λ − 1 ) ) . Then Γ w ( P t f ) ≤ e − 2 K λ t P t Γ w ( f ) \Gamma^w(P_tf)\le e^{-2K_\lambda t}P_t\Gamma^w(f) Γ w ( P t f ) ≤ e − 2 K λ t P t Γ w ( f ) f f f t ≥ 0 t\ge0 t ≥ 0
Proof. Put V = − N β S V=-N\beta S V = − N β S M = G A ′ M=G^{A'} M = G A ′ ∇ Δ g = Δ ˉ ∇ g + Ric ( ∇ g ) \nabla\Delta g=\bar\Delta\nabla g+\operatorname{Ric}(\nabla g) ∇Δ g = Δ ˉ ∇ g + Ric ( ∇ g ) g g g
1 2 L Γ w ( g ) − Γ w ( g , L g ) = ∑ e w e ( ∣ ∇ ∇ e g ∣ 2 + Ric G ( ∇ e g , ∇ e g ) ) + ∑ e , e ′ w e Hess V e e ′ ( ∇ e g , ∇ e ′ g ) , \frac{1}{2}L\Gamma^w(g)-\Gamma^w(g,Lg)
=\sum_e w_e\left(|\nabla\nabla_e g|^2+\operatorname{Ric}_G(\nabla_e g,\nabla_e g)\right)
+\sum_{e,e'}w_e\operatorname{Hess}V_{ee'}(\nabla_e g,\nabla_{e'}g), 2 1 L Γ w ( g ) − Γ w ( g , Lg ) = e ∑ w e ( ∣∇ ∇ e g ∣ 2 + Ric G ( ∇ e g , ∇ e g ) ) + e , e ′ ∑ w e Hess V e e ′ ( ∇ e g , ∇ e ′ g ) , where Γ w ( g , h ) \Gamma^w(g,h) Γ w ( g , h ) u e = ∇ e g u_e=\nabla_e g u e = ∇ e g v e = w e u e v_e=\sqrt{w_e}\,u_e v e = w e u e Hess V \operatorname{Hess}V Hess V
∑ e , e ′ w e Hess V e e ′ ( u e , u e ′ ) = Hess V ( v , v ) + ∑ e ≠ e ′ ( cosh λ ( d e − d e ′ ) 2 − 1 ) Hess V e e ′ ( v e , v e ′ ) , \sum_{e,e'}w_e\operatorname{Hess}V_{ee'}(u_e,u_{e'})=\operatorname{Hess}V(v,v)
+\sum_{e\ne e'}\left(\cosh\frac{\lambda(d_e-d_{e'})}{2}-1\right)\operatorname{Hess}V_{ee'}(v_e,v_{e'}), e , e ′ ∑ w e Hess V e e ′ ( u e , u e ′ ) = Hess V ( v , v ) + e = e ′ ∑ ( cosh 2 λ ( d e − d e ′ ) − 1 ) Hess V e e ′ ( v e , v e ′ ) , because w e + w e ′ 2 w e w e ′ = cosh λ ( d e − d e ′ ) 2 \frac{w_e+w_{e'}}{2\sqrt{w_ew_{e'}}}=\cosh\frac{\lambda(d_e-d_{e'})}{2} 2 w e w e ′ w e + w e ′ = cosh 2 λ ( d e − d e ′ ) − 29 2 N ∣ β ∣ ∣ v ∣ 2 -\frac{29}{2}N|\beta|\,|v|^2 − 2 29 N ∣ β ∣ ∣ v ∣ 2 cosh λ 2 − 1 \cosh\frac{\lambda}{2}-1 cosh 2 λ − 1 N ∣ β ∣ N|\beta| N ∣ β ∣ 2 ∣ v e ∣ ∣ v e ′ ∣ ≤ ∣ v e ∣ 2 + ∣ v e ′ ∣ 2 2|v_e||v_{e'}|\le|v_e|^2+|v_{e'}|^2 2∣ v e ∣∣ v e ′ ∣ ≤ ∣ v e ∣ 2 + ∣ v e ′ ∣ 2 12 ( cosh λ 2 − 1 ) N ∣ β ∣ ∣ v ∣ 2 12(\cosh\frac{\lambda}{2}-1)N|\beta|\,|v|^2 12 ( cosh 2 λ − 1 ) N ∣ β ∣ ∣ v ∣ 2 Ric G ≥ κ G \operatorname{Ric}_G\ge\kappa_G Ric G ≥ κ G ∣ v ∣ 2 = Γ w ( g ) |v|^2=\Gamma^w(g) ∣ v ∣ 2 = Γ w ( g )
1 2 L Γ w ( g ) − Γ w ( g , L g ) ≥ K λ Γ w ( g ) . \frac{1}{2}L\Gamma^w(g)-\Gamma^w(g,Lg)\ge K_\lambda\Gamma^w(g). 2 1 L Γ w ( g ) − Γ w ( g , Lg ) ≥ K λ Γ w ( g ) . For fixed t t t φ ( s ) = P s Γ w ( P t − s f ) \varphi(s)=P_s\Gamma^w(P_{t-s}f) φ ( s ) = P s Γ w ( P t − s f ) 0 ≤ s ≤ t 0\le s\le t 0 ≤ s ≤ t φ ′ ( s ) = P s ( L Γ w ( g s ) − 2 Γ w ( g s , L g s ) ) ≥ 2 K λ φ ( s ) \varphi'(s)=P_s(L\Gamma^w(g_s)-2\Gamma^w(g_s,Lg_s))\ge2K_\lambda\varphi(s) φ ′ ( s ) = P s ( L Γ w ( g s ) − 2 Γ w ( g s , L g s )) ≥ 2 K λ φ ( s ) g s = P t − s f g_s=P_{t-s}f g s = P t − s f φ ( 0 ) ≤ e − 2 K λ t φ ( t ) \varphi(0)\le e^{-2K_\lambda t}\varphi(t) φ ( 0 ) ≤ e − 2 K λ t φ ( t )
Lemma 25. Suppose K λ > 0 K_\lambda>0 K λ > 0 f f f g g g G A ′ G^{A'} G A ′ A f A_f A f B g B_g B g D D D A f A_f A f B g B_g B g A ′ A' A ′
∣ Cov μ ( f , g ) ∣ ≤ e − λ D / 2 K λ ∥ ∇ f ∥ ∞ ∑ e ∈ B g ∥ ∇ e g ∥ ∞ , |\operatorname{Cov}_\mu(f,g)|\le\frac{e^{-\lambda D/2}}{K_\lambda}\|\nabla f\|_\infty\sum_{e\in B_g}\|\nabla_e g\|_\infty, ∣ Cov μ ( f , g ) ∣ ≤ K λ e − λ D /2 ∥∇ f ∥ ∞ e ∈ B g ∑ ∥ ∇ e g ∥ ∞ , where $|\nabla f|_\infty^2=\sup\sum_e|\nabla_e f|^2.
Proof. Since K 0 ≥ K λ > 0 K_0 \ge K_{\lambda} > 0 K 0 ≥ K λ > 0 λ = 0 \lambda= 0 λ = 0 Γ ( P t f ) ≤ e − 2 K 0 t ∥ ∇ f ∥ ∞ 2 \Gamma(P_t f) \le e^{-2K_0t}\|\nabla f\|_\infty^2 Γ ( P t f ) ≤ e − 2 K 0 t ∥∇ f ∥ ∞ 2 P t f P_t f P t f μ ( f ) \mu(f) μ ( f ) M M M d d t μ ( g P t f ) = μ ( g L P t f ) = − μ ( Γ ( g , P t f ) ) \frac{d}{dt}\mu(gP_t f)=\mu(gLP_t f)=-\mu(\Gamma(g,P_t f)) d t d μ ( g P t f ) = μ ( g L P t f ) = − μ ( Γ ( g , P t f ))
Cov μ ( f , g ) = ∫ 0 ∞ μ ( Γ ( P t f , g ) ) d t . \operatorname{Cov}_{\mu}(f,g)=\int_0^\infty\mu(\Gamma(P_t f,g))\,dt. Cov μ ( f , g ) = ∫ 0 ∞ μ ( Γ ( P t f , g )) d t . Let d e d_e d e D D D e e e A f A_f A f ∣ d e − d e ′ ∣ ≤ 1 |d_e-d_{e'}|\le1 ∣ d e − d e ′ ∣ ≤ 1 w e = 1 w_e=1 w e = 1 A f A_f A f w e = e λ D w_e=e^{\lambda D} w e = e λ D B g B_g B g Γ w ( f ) ≤ ∥ ∇ f ∥ ∞ 2 \Gamma^w(f)\le\|\nabla f\|_\infty^2 Γ w ( f ) ≤ ∥∇ f ∥ ∞ 2 e ∈ B g e\in B_g e ∈ B g ∣ ∇ e P t f ∣ 2 ≤ w e − 1 Γ w ( P t f ) ≤ e λ D e − 2 K λ t ∥ ∇ f ∥ ∞ 2 |\nabla_eP_t f|^2\le w_e^{-1}\Gamma^w(P_t f)\le e^{\lambda D}e^{-2K_\lambda t}\|\nabla f\|_\infty^2 ∣ ∇ e P t f ∣ 2 ≤ w e − 1 Γ w ( P t f ) ≤ e λ D e − 2 K λ t ∥∇ f ∥ ∞ 2 ∣ Γ ( P t f , g ) ∣ ≤ e − λ D / 2 − K λ t ∥ ∇ f ∥ ∞ ∑ e ∈ B g ∥ ∇ e g ∥ ∞ |\Gamma(P_t f,g)|\le e^{-\lambda D/2-K_\lambda t}\|\nabla f\|_\infty\sum_{e\in B_g}\|\nabla_e g\|_\infty ∣Γ ( P t f , g ) ∣ ≤ e − λ D /2 − K λ t ∥∇ f ∥ ∞ ∑ e ∈ B g ∥ ∇ e g ∥ ∞ t t t □ \square □
Proof of Theorem 5 . Let λ ∗ > 0 \lambda^*>0 λ ∗ > 0 K λ ∗ = 0 K_{\lambda^*}=0 K λ ∗ = 0 λ ∗ 2 = m G ( β ) \frac{\lambda^*}{2}=m_G(\beta) 2 λ ∗ = m G ( β ) 0 < λ < λ ∗ 0<\lambda<\lambda^* 0 < λ < λ ∗ K λ > 0 K_\lambda>0 K λ > 0
Decay in boxes. Lemma 25 with S = S Λ S=S_\Lambda S = S Λ
∣ Cov μ Λ ( f , g ) ∣ ≤ C f , g e − λ D / 2 , |\operatorname{Cov}_{\mu_\Lambda}(f,g)|\le C_{f,g}e^{-\lambda D/2}, ∣ Cov μ Λ ( f , g ) ∣ ≤ C f , g e − λ D /2 , uniformly in the box.
Limit. For boxes Λ ⊂ Λ ′ \Lambda\subset\Lambda' Λ ⊂ Λ ′ S s = ( 1 − s ) S Λ + s S Λ ′ S_s=(1-s)S_\Lambda+sS_{\Lambda'} S s = ( 1 − s ) S Λ + s S Λ ′ Λ ′ \Lambda' Λ ′ s ∈ [ 0 , 1 ] s\in[0,1] s ∈ [ 0 , 1 ] μ s \mu_s μ s S s S_s S s μ 0 \mu_0 μ 0 μ Λ \mu_\Lambda μ Λ Λ \Lambda Λ μ 1 = μ Λ ′ \mu_1=\mu_{\Lambda'} μ 1 = μ Λ ′ f f f Λ \Lambda Λ
d d s μ s ( f ) = N β ∑ p ∈ P ( Λ ′ ) ∖ P ( Λ ) Cov μ s ( f , ReTr Q p F ) , \frac{d}{ds}\mu_s(f)=N\beta\sum_{p\in\mathcal{P}(\Lambda')\setminus\mathcal{P}(\Lambda)}\operatorname{Cov}_{\mu_s}(f,\operatorname{ReTr}Q_p^F), d s d μ s ( f ) = N β p ∈ P ( Λ ′ ) ∖ P ( Λ ) ∑ Cov μ s ( f , ReTr Q p F ) , and ∣ ∇ e ReTr Q p ∣ ≤ ∥ Q p ∥ H S = N |\nabla_e\operatorname{ReTr}Q_p|\le\|Q_p\|_{\mathrm{HS}}=\sqrt{N} ∣ ∇ e ReTr Q p ∣ ≤ ∥ Q p ∥ HS = N k k k T T T f f f k + 1 k+1 k + 1 T T T C T ( k + 1 ) 4 C_T(k+1)^4 C T ( k + 1 ) 4 P ( Λ ) \mathcal{P}(\Lambda) P ( Λ ) R − 1 R-1 R − 1 R R R T T T Λ \Lambda Λ
∣ μ Λ ′ F ( f ) − μ Λ F ( f ) ∣ ≤ C F R 4 e − λ R / 2 . |\mu_{\Lambda'}^F(f)-\mu_\Lambda^F(f)|\le C_FR^4e^{-\lambda R/2}. ∣ μ Λ ′ F ( f ) − μ Λ F ( f ) ∣ ≤ C F R 4 e − λ R /2 . The limit therefore exists along every increasing sequence of boxes, and two sequences are compared through their intersections, which are boxes. Continuous gauge-invariant O O O e t Δ O e^{t\Delta}O e t Δ O G E G^E G E Lemma 10 the limit ω ( O ) \omega(O) ω ( O ) W Λ ( O ) W_\Lambda(O) W Λ ( O )
Properties. Invariance and, for β ≥ 0 \beta\ge0 β ≥ 0 O O O e t Δ O e^{t\Delta}O e t Δ O Ric G ≥ 0 \operatorname{Ric}_G\ge0 Ric G ≥ 0 [4 Sect. 3.3] , so the bound of the first step passes to ω \omega ω t t t t − c t-c t − c m G ( β ) m_G(\beta) m G ( β ) □ \square □
Proof of the rest of Theorem 4 . By Lemma 21 and Table I , κ S U ( N ) = N 2 \kappa_{\mathrm{SU}(N)}=\frac{N}{2} κ SU ( N ) = 2 N κ S O ( N ) = N − 2 4 \kappa_{\mathrm{SO}(N)}=\frac{N-2}{4} κ SO ( N ) = 4 N − 2 S O ( 4 ) \mathrm{SO}(4) SO ( 4 ) s o ( 3 ) ⊕ s o ( 3 ) \mathfrak{so}(3)\oplus\mathfrak{so}(3) so ( 3 ) ⊕ so ( 3 ) R 4 ≃ C 2 ⊂ C 2 ⊕ C 2 \mathbb{R}^4\simeq\mathbb{C}^2\subset\mathbb{C}^2\oplus\mathbb{C}^2 R 4 ≃ C 2 ⊂ C 2 ⊕ C 2 1 2 \frac12 2 1 Theorem 4 are those of Theorem 5 . □ \square □
Remark 26. Let h F h_F h F ± Hess S 0 \pm\operatorname{Hess}S_0 ± Hess S 0 ∣ β ∣ < 1 / ( 2 h F ) |\beta|<1/(2h_F) ∣ β ∣ < 1/ ( 2 h F ) Hess S 0 \operatorname{Hess}S_0 Hess S 0 X F = 0 X_F=0 X F = 0 9 + 17 = 13.123 9+\sqrt{17}=13.123 9 + 17 = 13.123 9 + 17 ≤ h F ≤ 29 2 9+\sqrt{17}\le h_F\le\frac{29}{2} 9 + 17 ≤ h F ≤ 2 29 S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) h F = 13.4701 h_F=13.4701 h F = 13.4701 14.384 14.384 14.384 ∣ β ∣ = 1 29 |\beta|=\frac{1}{29} ∣ β ∣ = 29 1 1 26.25 \frac{1}{26.25} 26.25 1
SU(3) up to 3/4
In this section G = S U ( 3 ) G=\mathrm{SU}(3) G = SU ( 3 ) β ≥ 0 \beta\ge0 β ≥ 0 β W = 9 β \beta_W=9\beta β W = 9 β S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) d ( U , V ) = ∥ U − V ∥ H S d(U,V)=\|U-V\|_{\mathrm{HS}} d ( U , V ) = ∥ U − V ∥ HS ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y ) K = conv S U ( 3 ) K=\operatorname{conv}\mathrm{SU}(3) K = conv SU ( 3 ) A ( M ) = 1 2 ( M − M ∗ ) A(M)=\frac12(M-M^*) A ( M ) = 2 1 ( M − M ∗ ) P P P Ric = 3 2 \operatorname{Ric}=\frac32 Ric = 2 3 λ F = 8 3 \lambda_F=\frac83 λ F = 3 8 6 6 6
Reduction to one link
The conditional law of an active link is
ν b ( d U ) ∝ e ReTr ( U b ) d U , b = N β ∑ p ∋ e S p . \nu_b(dU)\propto e^{\operatorname{ReTr}(Ub)}\,dU,\qquad b=N\beta\sum_{p\ni e}S_p. ν b ( d U ) ∝ e ReTr ( U b ) d U , b = N β p ∋ e ∑ S p . where S p S_p S p e e e p p p S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) b ∈ r K b\in rK b ∈ r K r = 6 N β = 2 β W r=6N\beta=2\beta_W r = 6 N β = 2 β W h h h ϕ h ( U ) = ReTr ( U h ) \phi_h(U)=\operatorname{ReTr}(Uh) ϕ h ( U ) = ReTr ( U h )
L h ( b ) = sup { Cov ν b ( g , ϕ h ) : g 1 -Lipschitz for d } , L_h(b)=\sup\{\operatorname{Cov}_{\nu_b}(g,\phi_h):g\ 1\text{-Lipschitz for }d\}, L h ( b ) = sup { Cov ν b ( g , ϕ h ) : g 1 -Lipschitz for d } , Λ ( r ) = sup b ∈ r K sup ∥ h ∥ H S = 1 L h ( b ) . \Lambda(r)=\sup_{b\in rK}\ \sup_{\|h\|_{\mathrm{HS}}=1}L_h(b). Λ ( r ) = b ∈ r K sup ∥ h ∥ HS = 1 sup L h ( b ) . Changing a partner e ′ e' e ′ d ( Q e , Q e ′ ) d(Q_e,Q_{e'}) d ( Q e , Q e ′ ) b t b_t b t b b b b ′ b' b ′ r K rK r K d d t ν b t ( g ) = Cov ν b t ( g , ReTr ( U ( b ′ − b ) ) ) \frac{d}{dt}\nu_{b_t}(g)=\operatorname{Cov}_{\nu_{b_t}}(g,\operatorname{ReTr}(U(b'-b))) d t d ν b t ( g ) = Cov ν b t ( g , ReTr ( U ( b ′ − b ))) W 1 ( ν b , ν b ′ ) ≤ N β Λ ( r ) d ( Q e , Q e ′ ) . W_1(\nu_b,\nu_{b'})\le N\beta\Lambda(r)d(Q_e,Q_{e'}). W 1 ( ν b , ν b ′ ) ≤ N β Λ ( r ) d ( Q e , Q e ′ ) .
α ≤ 12 N β Λ ( r ) = 2 r Λ ( r ) = 4 β W Λ ( 2 β W ) . (8) \alpha\le12N\beta\Lambda(r)=2r\Lambda(r)=4\beta_W\Lambda(2\beta_W). \tag*{(8)} α ≤ 12 N β Λ ( r ) = 2 r Λ ( r ) = 4 β W Λ ( 2 β W ) . (8) Since K K K 0 0 0 I , ω , ω 2 I I,\omega,\omega^2I I , ω , ω 2 I ω 3 = 1 \omega^3=1 ω 3 = 1 r ′ K ⊂ r K r'K\subset rK r ′ K ⊂ r K r ′ ≤ r r'\le r r ′ ≤ r Λ \Lambda Λ
Lemma 27. In the left trivialisation, ∇ φ c ( U ) = − P A ( c U ) \nabla\varphi_c(U)=-PA(cU) ∇ φ c ( U ) = − P A ( c U ) V = φ b V=\varphi_b V = φ b Hess V ( X , X ) = ReTr ( U X 2 b ) \operatorname{Hess}V(X,X)=\operatorname{ReTr}(UX^2b) Hess V ( X , X ) = ReTr ( U X 2 b ) Δ φ c = − λ F φ c \Delta\varphi_c=-\lambda_F\varphi_c Δ φ c = − λ F φ c Proof. d d t Re Tr ( U e t X c ) ∣ t = 0 = Re Tr ( c U X ) = − ⟨ P A ( c U ) , X ⟩ \left.\frac{d}{dt}\operatorname{Re}\operatorname{Tr}(Ue^{tX}c)\right|_{t=0}=\operatorname{Re}\operatorname{Tr}(cUX)=-\langle PA(cU),X\rangle d t d Re Tr ( U e tX c ) t = 0 = Re Tr ( c U X ) = − ⟨ P A ( c U ) , X ⟩ X ∈ s u ( 3 ) X\in\mathfrak{su}(3) X ∈ su ( 3 ) ∇ X X = 0 \nabla_X X=0 ∇ X X = 0 X X X e t X e^{tX} e tX ϕ c \phi_c ϕ c λ F \lambda_F λ F
Lemma 28. The bottom of the spectrum of the rough Laplacian ∇ ∗ ∇ \nabla^*\nabla ∇ ∗ ∇ L 2 L^2 L 2 S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) μ 0 = 7 6 \mu_0=\frac{7}{6} μ 0 = 6 7
Proof. Trivialise one-forms on the left, so that a one-form is a function f : S U ( 3 ) → s u ( 3 ) f:\mathrm{SU}(3)\to\mathfrak{su}(3) f : SU ( 3 ) → su ( 3 ) X a X_a X a ∇ X a Y = 1 2 [ X a , Y ] \nabla_{X_a}Y=\frac{1}{2}[X_a,Y] ∇ X a Y = 2 1 [ X a , Y ] ∇ X a X a = 0 \nabla_{X_a}X_a=0 ∇ X a X a = 0 ∇ ∗ ∇ = − ∑ a ( ρ ( X a ) + 1 2 ad X a ) 2 \nabla^*\nabla=-\sum_a\left(\rho(X_a)+\frac{1}{2}\operatorname{ad}X_a\right)^2 ∇ ∗ ∇ = − ∑ a ( ρ ( X a ) + 2 1 ad X a ) 2 V π ⊗ V π ∗ ⊗ s u ( 3 ) V_\pi\otimes V_\pi^*\otimes\mathfrak{su}(3) V π ⊗ V π ∗ ⊗ su ( 3 ) L 2 ⊗ s u ( 3 ) L^2\otimes\mathfrak{su}(3) L 2 ⊗ su ( 3 ) A a = ρ π ( X a ) A_a=\rho_\pi(X_a) A a = ρ π ( X a ) B a = ad X a B_a=\operatorname{ad}X_a B a = ad X a T a = A a + B a T_a=A_a+B_a T a = A a + B a A a A_a A a B a B_a B a A a + 1 2 B a = 1 2 ( A a + T a ) A_a+\frac{1}{2}B_a=\frac{1}{2}(A_a+T_a) A a + 2 1 B a = 2 1 ( A a + T a ) A a T a + T a A a = A a 2 + T a 2 − B a 2 A_aT_a+T_aA_a=A_a^2+T_a^2-B_a^2 A a T a + T a A a = A a 2 + T a 2 − B a 2 σ ⊂ π ⊗ ad \sigma\subset\pi\otimes\operatorname{ad} σ ⊂ π ⊗ ad
∇ ∗ ∇ = 1 2 ( C π + C σ ) − 1 4 C a d = 1 2 ( C π + C σ ) − 3 2 . \nabla^*\nabla=\frac{1}{2}(C_\pi+C_\sigma)-\frac{1}{4}C_{\mathrm{ad}}=\frac{1}{2}(C_\pi+C_\sigma)-\frac{3}{2}. ∇ ∗ ∇ = 2 1 ( C π + C σ ) − 4 1 C ad = 2 1 ( C π + C σ ) − 2 3 . with C ( p , q ) = 2 3 ( p 2 + q 2 + p q + 3 p + 3 q ) C(p,q)=\frac{2}{3}(p^2+q^2+pq+3p+3q) C ( p , q ) = 3 2 ( p 2 + q 2 + pq + 3 p + 3 q ) ( p , q ) (p,q) ( p , q ) [34 ] . The components of ( p , q ) ⊗ ( 1 , 1 ) (p,q)\otimes(1,1) ( p , q ) ⊗ ( 1 , 1 ) ( p + 1 , q + 1 ) (p+1,q+1) ( p + 1 , q + 1 ) ( p + 2 , q − 1 ) (p+2,q-1) ( p + 2 , q − 1 ) ( p − 1 , q + 2 ) (p-1,q+2) ( p − 1 , q + 2 ) ( p + 1 , q − 2 ) (p+1,q-2) ( p + 1 , q − 2 ) ( p − 2 , q + 1 ) (p-2,q+1) ( p − 2 , q + 1 ) ( p − 1 , q − 1 ) (p-1,q-1) ( p − 1 , q − 1 ) ( p , q ) (p,q) ( p , q ) p , q p,q p , q π = ( 0 , 0 ) \pi=(0,0) π = ( 0 , 0 ) ( 1 , 1 ) (1,1) ( 1 , 1 ) 3 2 \frac{3}{2} 2 3 π = ( 1 , 0 ) \pi=(1,0) π = ( 1 , 0 ) ( 2 , 1 ) (2,1) ( 2 , 1 ) ( 0 , 2 ) (0,2) ( 0 , 2 ) ( 1 , 0 ) (1,0) ( 1 , 0 ) 31 6 \frac{31}{6} 6 31 7 6 \frac{7}{6} 6 7 7 6 \frac{7}{6} 6 7 ( 0 , 1 ) (0,1) ( 0 , 1 ) p + q ≥ 2 p+q\ge2 p + q ≥ 2 ( p − 1 , q − 1 ) (p-1,q-1) ( p − 1 , q − 1 ) ( p , q ) (p,q) ( p , q ) ( p − 2 , q + 1 ) (p-2,q+1) ( p − 2 , q + 1 ) ( p + 1 , q − 2 ) (p+1,q-2) ( p + 1 , q − 2 ) 1 3 ( 2 ( p 2 + q 2 + p q ) + 3 ( p + q ) − 3 ) − 3 2 ≥ 3 2 \frac{1}{3}\left(2(p^2+q^2+pq)+3(p+q)-3\right)-\frac{3}{2}\ge\frac{3}{2} 3 1 ( 2 ( p 2 + q 2 + pq ) + 3 ( p + q ) − 3 ) − 2 3 ≥ 2 3
The gap of the one-link measure
Lemma 29. For b ∈ r K b\in rK b ∈ r K L = Δ + ⟨ ∇ V , ∇ ⋅ ⟩ L=\Delta+\langle\nabla V,\nabla\cdot\rangle L = Δ + ⟨ ∇ V , ∇ ⋅ ⟩ V = ϕ b V=\phi_b V = ϕ b L 2 ( ν b ) L^2(\nu_b) L 2 ( ν b )
γ ( r ) = 39 64 μ 0 + 3 2 − r 2 = 283 128 − r 2 . \gamma(r)=\frac{39}{64}\mu_0+\frac{3}{2}-\frac{r}{2}=\frac{283}{128}-\frac{r}{2}. γ ( r ) = 64 39 μ 0 + 2 3 − 2 r = 128 283 − 2 r . Proof. Let − L f = λ f -Lf=\lambda f − L f = λ f λ > 0 \lambda>0 λ > 0 ω = d f \omega=df ω = df η = e V / 2 ω \eta=e^{V/2}\omega η = e V /2 ω L 2 ( d U ) L^2(dU) L 2 ( d U ) ν b \nu_b ν b λ ν b ∣ ω ∣ 2 = ν b ∣ ∇ ω ∣ 2 + ν b ( ( Ric − Hess V ) ( ω , ω ) ) \lambda\nu_b|\omega|^2=\nu_b|\nabla\omega|^2+\nu_b((\operatorname{Ric}-\operatorname{Hess}V)(\omega,\omega)) λ ν b ∣ ω ∣ 2 = ν b ∣∇ ω ∣ 2 + ν b (( Ric − Hess V ) ( ω , ω )) e V / 2 ∇ ω = ∇ η − 1 2 d V ⊗ η e^{V/2}\nabla\omega=\nabla\eta-\frac{1}{2}dV\otimes\eta e V /2 ∇ ω = ∇ η − 2 1 d V ⊗ η
λ ∥ η ∥ 2 = ∥ ∇ η − 1 2 d V ⊗ η ∥ 2 + ∫ ( Ric − Hess V ) ( η , η ) . (9) \lambda\|\eta\|^2=\left\|\nabla\eta-\frac{1}{2}dV\otimes\eta\right\|^2+\int(\operatorname{Ric}-\operatorname{Hess}V)(\eta,\eta). \tag*{(9)} λ ∥ η ∥ 2 = ∇ η − 2 1 d V ⊗ η 2 + ∫ ( Ric − Hess V ) ( η , η ) . (9) Put A = ∇ η A=\nabla\eta A = ∇ η B = 1 2 d V ⊗ η B=\frac{1}{2}dV\otimes\eta B = 2 1 d V ⊗ η
⟨ A , B ⟩ = 1 4 ⟨ ∇ V , ∇ ∣ η ∣ 2 ⟩ = λ F 4 ∫ V ∣ η ∣ 2 , \langle A,B\rangle=\frac{1}{4}\langle\nabla V,\nabla|\eta|^2\rangle=\frac{\lambda_F}{4}\int V|\eta|^2, ⟨ A , B ⟩ = 4 1 ⟨ ∇ V , ∇∣ η ∣ 2 ⟩ = 4 λ F ∫ V ∣ η ∣ 2 , because − Δ V = λ F V -\Delta V=\lambda_FV − Δ V = λ F V ∥ A − B ∥ 2 = ∥ 5 8 A − B ∥ 2 + 39 64 ∥ A ∥ 2 − 3 4 ⟨ A , B ⟩ \|A-B\|^2=\|\frac{5}{8}A-B\|^2+\frac{39}{64}\|A\|^2-\frac{3}{4}\langle A,B\rangle ∥ A − B ∥ 2 = ∥ 8 5 A − B ∥ 2 + 64 39 ∥ A ∥ 2 − 4 3 ⟨ A , B ⟩
∥ A − B ∥ 2 ≥ 39 64 μ 0 ∥ η ∥ 2 − 1 2 ∫ V ∣ η ∣ 2 \|A-B\|^2\ge\frac{39}{64}\mu_0\|\eta\|^2-\frac{1}{2}\int V|\eta|^2 ∥ A − B ∥ 2 ≥ 64 39 μ 0 ∥ η ∥ 2 − 2 1 ∫ V ∣ η ∣ 2 by Lemma 28. With Ric = 3 2 \operatorname{Ric}=\frac{3}{2} Ric = 2 3 (9) gives λ ∥ η ∥ 2 ≥ ( 39 64 μ 0 + 3 2 ) ∥ η ∥ 2 − ∫ ( Hess V ( η , η ) + 1 2 V ∣ η ∣ 2 ) \lambda\|\eta\|^2\ge\left(\frac{39}{64}\mu_0+\frac{3}{2}\right)\|\eta\|^2-\int\left(\operatorname{Hess}V(\eta,\eta)+\frac{1}{2}V|\eta|^2\right) λ ∥ η ∥ 2 ≥ ( 64 39 μ 0 + 2 3 ) ∥ η ∥ 2 − ∫ ( Hess V ( η , η ) + 2 1 V ∣ η ∣ 2 )
Re Tr ( M X 2 ) + 1 2 Re Tr M ≤ r 2 ∣ X ∣ 2 , M = b U ∈ r K , (10) \operatorname{Re}\operatorname{Tr}(MX^2)+\frac{1}{2}\operatorname{Re}\operatorname{Tr}M\le\frac{r}{2}|X|^2,\qquad M=bU\in rK, \tag*{(10)} Re Tr ( M X 2 ) + 2 1 Re Tr M ≤ 2 r ∣ X ∣ 2 , M = b U ∈ r K , (10) since Hess V ( X , X ) = Re Tr ( M X 2 ) \operatorname{Hess}V(X,X)=\operatorname{Re}\operatorname{Tr}(MX^2) Hess V ( X , X ) = Re Tr ( M X 2 ) V = Re Tr M V=\operatorname{Re}\operatorname{Tr}M V = Re Tr M ∣ X ∣ = 1 |X|=1 ∣ X ∣ = 1 M M M M = r W M=rW M = r W W ∈ S U ( 3 ) W\in\mathrm{SU}(3) W ∈ SU ( 3 ) W = diag ( e i θ k ) W=\operatorname{diag}(e^{i\theta_k}) W = diag ( e i θ k ) ∑ k θ k = 0 \sum_k\theta_k=0 ∑ k θ k = 0 c k = cos θ k c_k=\cos\theta_k c k = cos θ k X = i H X=iH X = i H H H H Tr H 2 = 1 \operatorname{Tr}H^2=1 Tr H 2 = 1 d k d_k d k H j k H_{jk} H j k
Re Tr ( W X 2 ) = − ∑ k c k d k 2 − ∑ j < k ( c j + c k ) ∣ H j k ∣ 2 , \operatorname{Re}\operatorname{Tr}(WX^2)=-\sum_k c_kd_k^2-\sum_{j<k}(c_j+c_k)|H_{jk}|^2, Re Tr ( W X 2 ) = − k ∑ c k d k 2 − j < k ∑ ( c j + c k ) ∣ H j k ∣ 2 , with ∑ k d k 2 + 2 ∑ j < k ∣ H j k ∣ 2 = 1 \sum_kd_k^2+2\sum_{j<k}|H_{jk}|^2=1 ∑ k d k 2 + 2 ∑ j < k ∣ H j k ∣ 2 = 1 q o f f = max j < k 1 2 ( c j + c k ) q_{\mathrm{off}}=\max_{j<k}\frac{1}{2}(c_j+c_k) q off = max j < k 2 1 ( c j + c k ) q d i a g = max { − ∑ k c k d k 2 : ∑ k d k = 0 , ∑ k d k 2 = 1 } q_{\mathrm{diag}}=\max\{-\sum_kc_kd_k^2:\sum_kd_k=0,\sum_kd_k^2=1\} q diag = max { − ∑ k c k d k 2 : ∑ k d k = 0 , ∑ k d k 2 = 1 } l l l q o f f + 1 2 ∑ k c k = 1 2 c l ≤ 1 2 q_{\mathrm{off}}+\frac{1}{2}\sum_kc_k=\frac{1}{2}c_l\le\frac{1}{2} q off + 2 1 ∑ k c k = 2 1 c l ≤ 2 1 q d i a g q_{\mathrm{diag}} q diag diag ( c ) \operatorname{diag}(c) diag ( c ) ∑ k d k = 0 \sum_kd_k=0 ∑ k d k = 0 ∑ k ( c k − μ ) − 1 = 0 \sum_k(c_k-\mu)^{-1}=0 ∑ k ( c k − μ ) − 1 = 0 3 μ 2 − 2 μ ∑ l c l + ∑ l < m c l c m = 0 3\mu^2-2\mu\sum_lc_l+\sum_{l<m}c_lc_m=0 3 μ 2 − 2 μ ∑ l c l + ∑ l < m c l c m = 0 e 1 , e 2 e_1,e_2 e 1 , e 2 c k c_k c k
q d i a g = 1 3 ( e 1 2 − 3 e 2 − e 1 ) . q_{\mathrm{diag}}=\frac{1}{3}\left(\sqrt{e_1^2-3e_2}-e_1\right). q diag = 3 1 ( e 1 2 − 3 e 2 − e 1 ) . The inequality q d i a g + 1 2 e 1 ≤ 1 2 q_{\mathrm{diag}}+\frac{1}{2}e_1\le\frac{1}{2} q diag + 2 1 e 1 ≤ 2 1 e 1 2 − 3 e 2 ≤ 1 2 ( 3 − e 1 ) \sqrt{e_1^2-3e_2}\le\frac{1}{2}(3-e_1) e 1 2 − 3 e 2 ≤ 2 1 ( 3 − e 1 ) e 1 2 − 4 e 2 + 2 e 1 − 3 ≤ 0 e_1^2-4e_2+2e_1-3\le0 e 1 2 − 4 e 2 + 2 e 1 − 3 ≤ 0 ∑ k θ k = 0 \sum_k\theta_k=0 ∑ k θ k = 0 ∑ k c k 2 = 1 + 2 c 1 c 2 c 3 \sum_kc_k^2=1+2c_1c_2c_3 ∑ k c k 2 = 1 + 2 c 1 c 2 c 3 − 2 ( 1 − c 1 ) ( 1 − c 2 ) ( 1 − c 3 ) ≤ 0 -2(1-c_1)(1-c_2)(1-c_3)\le0 − 2 ( 1 − c 1 ) ( 1 − c 2 ) ( 1 − c 3 ) ≤ 0
The weight 5 8 \frac{5}{8} 8 5 1 2 \frac{1}{2} 2 1 V V V (10) is an equality at W = I W=I W = I r = 3 2 r=\frac{3}{2} r = 2 3 γ = 187 128 \gamma=\frac{187}{128} γ = 128 187
A corrector of degree two
Lemma 30. For b ∈ r K b\in rK b ∈ r K ∥ h ∥ = 1 \|h\|=1 ∥ h ∥ = 1 u u u S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
L h ( b ) ≤ ν b ∣ ∇ u ∣ 2 + Var ν b ( ϕ h + L u ) / γ ( r ) . L_h(b)\le\sqrt{\nu_b|\nabla u|^2}+\sqrt{\operatorname{Var}_{\nu_b}(\phi_h+Lu)/\gamma(r)}. L h ( b ) ≤ ν b ∣∇ u ∣ 2 + Var ν b ( ϕ h + Lu ) / γ ( r ) . Proof. Let − L ψ = ϕ h − ν b ϕ h -L\psi=\phi_h-\nu_b\phi_h − L ψ = ϕ h − ν b ϕ h 1 1 1 g g g ∣ ∇ g ∣ ≤ 1 |\nabla g|\le1 ∣∇ g ∣ ≤ 1 d d d Cov ( g , ϕ h ) = ν b ⟨ ∇ g , ∇ ψ ⟩ ≤ ν b ∣ ∇ ψ ∣ \operatorname{Cov}(g,\phi_h)=\nu_b\langle\nabla g,\nabla\psi\rangle\le\nu_b|\nabla\psi| Cov ( g , ϕ h ) = ν b ⟨ ∇ g , ∇ ψ ⟩ ≤ ν b ∣∇ ψ ∣ ψ = u + χ \psi=u+\chi ψ = u + χ ν b ( L u ) = 0 \nu_b(Lu)=0 ν b ( Lu ) = 0 χ \chi χ − L χ = X − ν b X -L\chi=X-\nu_bX − L χ = X − ν b X X = ϕ h + L u X=\phi_h+Lu X = ϕ h + Lu
ν b ∣ ∇ χ ∣ 2 = Cov ( χ , X ) ≤ Var χ Var X ≤ ν b ∣ ∇ χ ∣ 2 Var X γ \nu_b|\nabla\chi|^2=\operatorname{Cov}(\chi,X)\le\sqrt{\operatorname{Var}\chi\operatorname{Var}X}\le\sqrt{\frac{\nu_b|\nabla\chi|^2\operatorname{Var}X}{\gamma}} ν b ∣∇ χ ∣ 2 = Cov ( χ , X ) ≤ Var χ Var X ≤ γ ν b ∣∇ χ ∣ 2 Var X by Lemma 29. Hence ν b ∣ ∇ χ ∣ 2 ≤ Var X / γ \nu_b|\nabla\chi|^2\le\operatorname{Var}X/\gamma ν b ∣∇ χ ∣ 2 ≤ Var X / γ ν b ∣ ∇ ψ ∣ ≤ ν b ∣ ∇ u ∣ 2 + ν b ∣ ∇ χ ∣ 2 \nu_b|\nabla\psi|\le\sqrt{\nu_b|\nabla u|^2}+\sqrt{\nu_b|\nabla\chi|^2} ν b ∣∇ ψ ∣ ≤ ν b ∣∇ u ∣ 2 + ν b ∣∇ χ ∣ 2
The best u u u ψ \psi ψ u = s φ h u=s\varphi_h u = s φ h φ h + L u = ( 1 − s λ F ) φ h + s Γ \varphi_h+Lu=(1-s\lambda_F)\varphi_h+s\Gamma φ h + Lu = ( 1 − s λ F ) φ h + s Γ Γ = ⟨ ∇ V , ∇ φ h ⟩ \Gamma=\langle\nabla V,\nabla\varphi_h\rangle Γ = ⟨ ∇ V , ∇ φ h ⟩ s = 1 / λ F s=1/\lambda_F s = 1/ λ F φ h \varphi_h φ h β W = 45 64 \beta_W=\frac{45}{64} β W = 64 45 § 2.6 ). We take u u u
Let P 2 \mathcal{P}_2 P 2 u i j u_{ij} u ij U U U u ˉ i j \bar{u}_{ij} u ˉ ij U ˉ \bar{U} U ˉ ( B k ) (B_k) ( B k ) 1 2 ( m + m ˉ ) \frac{1}{2}(m+\bar{m}) 2 1 ( m + m ˉ ) 1 2 i ( m − m ˉ ) \frac{1}{2i}(m-\bar{m}) 2 i 1 ( m − m ˉ ) { m , m ˉ } \{m,\bar{m}\} { m , m ˉ } m ≠ m ˉ m\ne\bar{m} m = m ˉ m m m m = m ˉ m=\bar{m} m = m ˉ h ∈ C 3 × 3 h\in\mathbb{C}^{3\times3} h ∈ C 3 × 3 R 18 \mathbb{R}^{18} R 18 189 × 18 189\times18 189 × 18 C C C q h = ∑ k ∑ m = 1 18 C k m h m B k ∈ P 2 q_h=\sum_{k}\sum_{m=1}^{18}C_{km}h_mB_k\in\mathcal{P}_2 q h = ∑ k ∑ m = 1 18 C k m h m B k ∈ P 2
q h = ∑ k ∑ m = 1 18 C k m h m B k ∈ P 2 , q_h=\sum_k\sum_{m=1}^{18} C_{km}h_mB_k\in\mathcal{P}_2, q h = k ∑ m = 1 ∑ 18 C k m h m B k ∈ P 2 , linear in h h h b = diag ( a 1 , a 2 , x 3 + i y 3 ) b=\operatorname{diag}(a_1,a_2,x_3+iy_3) b = diag ( a 1 , a 2 , x 3 + i y 3 ) V = ∑ k b k φ F k V=\sum_k b_k\varphi_{F_k} V = ∑ k b k φ F k ( b k ) = ( a 1 , a 2 , x 3 , y 3 ) (b_k)=(a_1,a_2,x_3,y_3) ( b k ) = ( a 1 , a 2 , x 3 , y 3 ) F = ( e 11 , e 22 , e 33 , i e 33 ) F=(e_{11},e_{22},e_{33},ie_{33}) F = ( e 11 , e 22 , e 33 , i e 33 )
X h = φ h + L q h = φ h + Δ q h + ∑ k b k ⟨ ∇ φ F k , ∇ q h ⟩ (11) X_h=\varphi_h+Lq_h=\varphi_h+\Delta q_h+\sum_k b_k\langle\nabla\varphi_{F_k},\nabla q_h\rangle \tag*{(11)} X h = φ h + L q h = φ h + Δ q h + k ∑ b k ⟨ ∇ φ F k , ∇ q h ⟩ (11) is a polynomial of degree at most three, affine in b b b ∇ q h \nabla q_h ∇ q h b b b h ↦ ν b ∣ ∇ q h ∣ 2 h\mapsto\nu_b\lvert\nabla q_h\rvert^2 h ↦ ν b ∣ ∇ q h ∣ 2 h ↦ ν b [ ( X h − κ ) 2 ] h\mapsto\nu_b[(X_h-\kappa)^2] h ↦ ν b [( X h − κ ) 2 ] ν b \nu_b ν b
To compute Δ \Delta Δ ⟨ ∇ f , ∇ g ⟩ \langle\nabla f,\nabla g\rangle ⟨ ∇ f , ∇ g ⟩ s u ( 3 ) \mathfrak{su}(3) su ( 3 )
e j k − e k j , i ( e j k + e k j ) ( j < k ) , i ( e 11 − e 22 ) , i ( e 11 + e 22 − 2 e 33 ) , e_{jk}-e_{kj},\qquad i(e_{jk}+e_{kj})\quad(j<k),\qquad i(e_{11}-e_{22}),\qquad i(e_{11}+e_{22}-2e_{33}), e j k − e k j , i ( e j k + e k j ) ( j < k ) , i ( e 11 − e 22 ) , i ( e 11 + e 22 − 2 e 33 ) , acting as left-invariant fields X a f ( U ) = d d t f ( U e t X a ) ∣ t = 0 X_af(U)=\left.\frac{d}{dt}f(Ue^{tX_a})\right|_{t=0} X a f ( U ) = d t d f ( U e t X a ) t = 0 w a = 1 2 w_a=\frac{1}{2} w a = 2 1 1 6 \frac{1}{6} 6 1 ∑ a w a X a ⊗ X a \sum_aw_aX_a\otimes X_a ∑ a w a X a ⊗ X a ⟨ X , Y ⟩ = − Tr ( X Y ) \langle X,Y\rangle=-\operatorname{Tr}(XY) ⟨ X , Y ⟩ = − Tr ( X Y )
⟨ ∇ f , ∇ g ⟩ = ∑ a w a X a f X a g , Δ = ∑ a w a X a 2 . \langle\nabla f,\nabla g\rangle=\sum_a w_aX_afX_ag,\qquad\Delta=\sum_a w_aX_a^2. ⟨ ∇ f , ∇ g ⟩ = a ∑ w a X a f X a g , Δ = a ∑ w a X a 2 . The fields map monomials to combinations of monomials of the same degree with coefficients in Z [ i ] \mathbb{Z}[i] Z [ i ] Δ φ h = − 8 3 φ h \Delta\varphi_h=-\frac{8}{3}\varphi_h Δ φ h = − 3 8 φ h
Orbits of staple sums
Lemma 31. Let V , W ∈ S U ( 3 ) V,W\in\mathrm{SU}(3) V , W ∈ SU ( 3 )
L h ( V b W ) = L V − 1 h W − 1 ( b ) L_h(VbW)=L_{V^{-1}hW^{-1}}(b) L h ( V bW ) = L V − 1 h W − 1 ( b ) L h ( b ˉ ) = L h ˉ ( b ) L_h(\bar{b})=L_{\bar{h}}(b) L h ( b ˉ ) = L h ˉ ( b )
Every b ∈ C 3 × 3 b\in\mathbb{C}^{3\times3} b ∈ C 3 × 3 diag ( a 1 , a 2 , z 3 ) \operatorname{diag}(a_1,a_2,z_3) diag ( a 1 , a 2 , z 3 ) a 1 ≥ a 2 ≥ ∣ z 3 ∣ a_1\ge a_2\ge\lvert z_3\rvert a 1 ≥ a 2 ≥ ∣ z 3 ∣ Im z 3 ≥ 0 \operatorname{Im}z_3\ge0 Im z 3 ≥ 0
A diagonal matrix lies in K K K T T T S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
Proof. (i) If U U U ν V b W \nu_{VbW} ν V bW W U V WUV W U V ν b \nu_b ν b d d d φ h ( U ) = φ V − 1 h W − 1 ( W U V ) \varphi_h(U)=\varphi_{V^{-1}hW^{-1}}(WUV) φ h ( U ) = φ V − 1 h W − 1 ( W U V ) b ˉ \bar{b} b ˉ U ↦ U ˉ U\mapsto\bar{U} U ↦ U ˉ b = V 1 Σ W 1 b=V_1\Sigma W_1 b = V 1 Σ W 1 V 1 , W 1 V_1,W_1 V 1 , W 1 Im z 3 ≥ 0 \operatorname{Im}z_3\ge0 Im z 3 ≥ 0 c c c U U U Re Tr ( c U ) \operatorname{Re}\operatorname{Tr}(cU) Re Tr ( c U ) S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) s u ( 3 ) \mathfrak{su}(3) su ( 3 ) c U cU c U i λ I i\lambda I iλ I c U = H + i λ I cU=H+i\lambda I c U = H + iλ I H H H H 2 + λ 2 I = c c ∗ H^2+\lambda^2I=cc^* H 2 + λ 2 I = c c ∗ H H H U ∈ T U\in T U ∈ T max U ∈ S U ( 3 ) Re Tr ( c U ) = max U ∈ T Re Tr ( c U ) \max_{U\in\mathrm{SU}(3)}\operatorname{Re}\operatorname{Tr}(cU)=\max_{U\in T}\operatorname{Re}\operatorname{Tr}(cU) max U ∈ SU ( 3 ) Re Tr ( c U ) = max U ∈ T Re Tr ( c U ) c c c c c c conv T \operatorname{conv}T conv T K K K □ \square □
Since every b ∈ r K b\in rK b ∈ r K r r r
Λ ( r ) = sup ∥ h ∥ = 1 { sup b ∈ R r L h ( b ) } , \Lambda(r)=\sup_{\lVert h\rVert=1}\left\{\sup_{b\in\mathcal{R}_r}L_h(b)\right\}, Λ ( r ) = ∥ h ∥ = 1 sup { b ∈ R r sup L h ( b ) } , where R r \mathcal{R}_r R r diag ( a 1 , a 2 , x 3 + i y 3 ) ∈ r conv T \operatorname{diag}(a_1,a_2,x_3+iy_3)\in r\operatorname{conv}T diag ( a 1 , a 2 , x 3 + i y 3 ) ∈ r conv T a 1 ≥ a 2 ≥ ∣ x 3 + i y 3 ∣ a_1\ge a_2\ge\lvert x_3+iy_3\rvert a 1 ≥ a 2 ≥ ∣ x 3 + i y 3 ∣ y 3 ≥ 0 y_3\ge0 y 3 ≥ 0 [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^2\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] ( a 1 , a 2 , x 3 , y 3 ) (a_1,a_2,x_3,y_3) ( a 1 , a 2 , x 3 , y 3 )
Exact moments and interpolation
Lemma 32. For D , D ′ ∈ C 3 × 3 D,D'\in\mathbb{C}^{3\times3} D , D ′ ∈ C 3 × 3 E H \mathbb{E}_H E H S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
Z ( D , D ′ ) = E H exp ( 1 2 Tr ( U D ) + 1 2 Tr ( U ˉ D ′ ) ) . Z(D,D')=\mathbb{E}_H\exp\left(\frac{1}{2}\operatorname{Tr}(UD)+\frac{1}{2}\operatorname{Tr}(\bar{U}D')\right). Z ( D , D ′ ) = E H exp ( 2 1 Tr ( U D ) + 2 1 Tr ( U ˉ D ′ ) ) . Then
Z ( D , D ′ ) = ∑ p ≥ q ≥ 0 s ( p , q ) ( D D ′ T ) G p , q ( det D ) G p , q ( det D ′ ) d p , q , Z(D,D')=\sum_{p\ge q\ge0}\frac{s_{(p,q)}(DD'^{\mathsf T})G_{p,q}(\det D)G_{p,q}(\det D')}{d_{p,q}}, Z ( D , D ′ ) = p ≥ q ≥ 0 ∑ d p , q s ( p , q ) ( D D ′ T ) G p , q ( det D ) G p , q ( det D ′ ) , with G p , q ( z ) = ∑ k ≥ 0 c ( p + k , q + k , k ) z k G_{p,q}(z)=\sum_{k\ge0}c_{(p+k,q+k,k)}z^k G p , q ( z ) = ∑ k ≥ 0 c ( p + k , q + k , k ) z k c λ = 2 − ∣ λ ∣ / ∏ hooks ( λ ) c_\lambda=2^{-\lvert\lambda\rvert}/\prod\operatorname{hooks}(\lambda) c λ = 2 − ∣ λ ∣ / ∏ hooks ( λ ) d p , q = 1 2 ( p − q + 1 ) ( q + 1 ) ( p + 2 ) d_{p,q}=\frac{1}{2}(p-q+1)(q+1)(p+2) d p , q = 2 1 ( p − q + 1 ) ( q + 1 ) ( p + 2 ) s ( p , q ) s_{(p,q)} s ( p , q ) j j j U U U k k k U ˉ \bar{U} U ˉ
E H [ u α u ˉ β ] Re Tr ( U b ) = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ˉ ) , \mathbb{E}_H[u^\alpha\bar{u}^\beta]\operatorname{Re}\operatorname{Tr}(Ub)=2^{\lvert\alpha\rvert+\lvert\beta\rvert}\partial_D^\alpha\partial_{D'}^\beta Z(b,\bar{b}), E H [ u α u ˉ β ] Re Tr ( U b ) = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ˉ ) , the entry u i j u_{ij} u ij D j i D_{ji} D j i
Proof. We use ( Tr X ) n = ∑ λ f λ s λ ( X ) (\operatorname{Tr}X)^n=\sum_\lambda f^\lambda s_\lambda(X) ( Tr X ) n = ∑ λ f λ s λ ( X ) f λ f^\lambda f λ λ \lambda λ [71 Cor. 7.21.6] . Expanding the exponential gives
e 1 2 Tr X = ∑ n ≥ 0 ( Tr X ) n 2 n n ! = ∑ λ c λ s λ ( X ) , e^{\frac{1}{2}\operatorname{Tr}X}=\sum_{n\ge0}\frac{(\operatorname{Tr}X)^n}{2^n n!}=\sum_\lambda c_\lambda s_\lambda(X), e 2 1 Tr X = n ≥ 0 ∑ 2 n n ! ( Tr X ) n = λ ∑ c λ s λ ( X ) , the sum over partitions with at most three parts. For U ∈ S U ( 3 ) U\in\mathrm{SU}(3) U ∈ SU ( 3 ) s λ + ( k , k , k ) ( U D ) = ( det D ) k s λ ( U D ) s_{\lambda+(k,k,k)}(UD)=(\det D)^ks_\lambda(UD) s λ + ( k , k , k ) ( U D ) = ( det D ) k s λ ( U D )
e 1 2 Tr ( U D ) = ∑ p ≥ q G p , q ( det D ) χ p , q ( U D ) , e^{\frac{1}{2}\operatorname{Tr}(UD)}=\sum_{p\ge q}G_{p,q}(\det D)\chi_{p,q}(UD), e 2 1 Tr ( U D ) = p ≥ q ∑ G p , q ( det D ) χ p , q ( U D ) , where χ p , q \chi_{p,q} χ p , q ( p , q , 0 ) (p,q,0) ( p , q , 0 ) d p , q d_{p,q} d p , q U ˉ = ( U − 1 ) T \bar{U}=(U^{-1})^{\mathsf T} U ˉ = ( U − 1 ) T χ p , q ( U D ˉ ′ ) = χ p , q ( U − 1 D ′ T ) \chi_{p,q}(U\bar{D}')=\chi_{p,q}(U^{-1}D'^{\mathsf T}) χ p , q ( U D ˉ ′ ) = χ p , q ( U − 1 D ′ T ) [22 Ch. 2] ,
E H [ χ p ( U A ) χ p ′ ( U − 1 B ) ] = δ p p ′ χ p ( A B ) d p , \mathbb{E}_H\left[\chi_p(UA)\chi_{p'}(U^{-1}B)\right]=\delta_{pp'}\frac{\chi_p(AB)}{d_p}, E H [ χ p ( U A ) χ p ′ ( U − 1 B ) ] = δ p p ′ d p χ p ( A B ) , gives the expansion. The bound below gives absolute convergence. Finally Re Tr ( U b ) = 1 2 Tr ( U b ) + 1 2 Tr ( U b ‾ ) \operatorname{Re}\operatorname{Tr}(U_b)=\frac{1}{2}\operatorname{Tr}(U_b)+\frac{1}{2}\operatorname{Tr}(\overline{U_b}) Re Tr ( U b ) = 2 1 Tr ( U b ) + 2 1 Tr ( U b )
The computation keeps p ≤ 34 p\le34 p ≤ 34 k ≤ 11 k\le11 k ≤ 11 ∣ s λ ( X ) ∣ ≤ dim ( λ ) ∥ X ∥ ∣ λ ∣ |s_\lambda(X)|\le\dim(\lambda)\|X\|^{|\lambda|} ∣ s λ ( X ) ∣ ≤ dim ( λ ) ∥ X ∥ ∣ λ ∣ ∑ λ ⊢ n f λ dim λ = 3 n \sum_{\lambda\vdash n} f^\lambda\dim\lambda=3^n ∑ λ ⊢ n f λ dim λ = 3 n ∑ n > 34 ( 2 3 ∥ D ∥ ) n / n ! \sum_{n>34}(\frac{2}{3}\|D\|)^n/n! ∑ n > 34 ( 3 2 ∥ D ∥ ) n / n ! b b b
Lemma 33 (Interpolation on a box). Let Q = ∏ k [ l k , l k + w k ] Q=\prod_k[l_k,l_k+w_k] Q = ∏ k [ l k , l k + w k ] ( a 1 , a 2 , x 3 , y 3 ) (a_1,a_2,x_3,y_3) ( a 1 , a 2 , x 3 , y 3 ) b i b_i b i ω i ( b ) ≥ 0 \omega_i(b)\ge0 ω i ( b ) ≥ 0 ∑ i ω i ( b ) = 1 \sum_i\omega_i(b)=1 ∑ i ω i ( b ) = 1 ς = w 1 2 + w 2 2 + max ( w 3 2 , w 4 2 ) \varsigma=w_1^2+w_2^2+\max(w_3^2,w_4^2) ς = w 1 2 + w 2 2 + max ( w 3 2 , w 4 2 ) b ∈ Q b\in Q b ∈ Q
(i) Z ( b ) ≥ e − ς / 8 ∑ i ω i ( b ) Z ( b i ) Z(b)\ge e^{-\varsigma/8}\sum_i\omega_i(b)Z(b_i) Z ( b ) ≥ e − ς /8 ∑ i ω i ( b ) Z ( b i )
(ii) ν b ( f ) ≤ e ς / 8 max i ν b i ( f ) \nu_b(f)\le e^{\varsigma/8}\max_i\nu_{b_i}(f) ν b ( f ) ≤ e ς /8 max i ν b i ( f ) f ≥ 0 f\ge0 f ≥ 0 b b b
(iii) if Y b = ( A + ∑ k ( b k − β k ) B k ) 2 Y_b=(\mathcal{A}+\sum_k(b_k-\beta_k)\mathcal{B}_k)^2 Y b = ( A + ∑ k ( b k − β k ) B k ) 2 A , B k \mathcal{A},\mathcal{B}_k A , B k U U U β \beta β
ν b ( Y b ) ≤ e ς / 8 ( max i ν b i ( Y b i ) + ∑ k w k 2 4 max i Z ( b i ) ν b i ( B k 2 ) min i Z ( b i ) ) . \nu_b(Y_b)\le e^{\varsigma/8}\left(\max_i\nu_{b_i}(Y_{b_i})+\sum_k\frac{w_k^2}{4}\frac{\max_i Z(b_i)\nu_{b_i}(\mathcal{B}_k^2)}{\min_i Z(b_i)}\right). ν b ( Y b ) ≤ e ς /8 ( i max ν b i ( Y b i ) + k ∑ 4 w k 2 min i Z ( b i ) max i Z ( b i ) ν b i ( B k 2 ) ) . Proof. Write E ( b ) = e Re Tr ( U b ) = ∏ k e c k b k E(b)=e^{\operatorname{Re}\operatorname{Tr}(Ub)}=\prod_k e^{c_kb_k} E ( b ) = e Re Tr ( U b ) = ∏ k e c k b k
c = ( Re u 11 , Re u 22 , Re u 33 , − Im u 33 ) , c=(\operatorname{Re}u_{11},\operatorname{Re}u_{22},\operatorname{Re}u_{33},-\operatorname{Im}u_{33}), c = ( Re u 11 , Re u 22 , Re u 33 , − Im u 33 ) , so that ∣ c k ∣ ≤ 1 |c_k|\le1 ∣ c k ∣ ≤ 1 c 3 2 + c 4 2 ≤ 1 c_3^2+c_4^2\le1 c 3 2 + c 4 2 ≤ 1 e c k x e^{c_kx} e c k x [ l k , l k + w k ] [l_k,l_k+w_k] [ l k , l k + w k ] [45 ] for a two-point variable, below e c k x e c k 2 w k 2 / 8 e^{c_kx}e^{c_k^2w_k^2/8} e c k x e c k 2 w k 2 /8
E ( b ) ≤ ∑ i ω i ( b ) E ( b i ) ≤ E ( b ) e ∑ k c k 2 w k 2 / 8 ≤ E ( b ) e ς / 8 . (12) E(b)\le\sum_i\omega_i(b)E(b_i)\le E(b)e^{\sum_k c_k^2w_k^2/8}\le E(b)e^{\varsigma/8}. \tag*{(12)} E ( b ) ≤ i ∑ ω i ( b ) E ( b i ) ≤ E ( b ) e ∑ k c k 2 w k 2 /8 ≤ E ( b ) e ς /8 . (12) Integrating the right inequality gives (i), and (ii) follows from the left one and (i). For (iii) let F ( b ) = Y b E ( b ) \mathcal{F}(b)=Y_bE(b) F ( b ) = Y b E ( b ) I k I_k I k k k k F − I 1 I 2 I 3 I 4 F = ∑ k I 1 ⋯ I k − 1 ( F − I k F ) \mathcal{F}-I_1I_2I_3I_4\mathcal{F}=\sum_k I_1\cdots I_{k-1}(\mathcal{F}-I_k\mathcal{F}) F − I 1 I 2 I 3 I 4 F = ∑ k I 1 ⋯ I k − 1 ( F − I k F ) k k k F \mathcal{F} F ℓ 2 e c k x \ell^2e^{c_kx} ℓ 2 e c k x ℓ \ell ℓ B k \mathcal{B}_k B k ∂ k 2 F = ( ( c k ℓ + 2 B k ) 2 − 2 B k 2 ) E ≥ − 2 B k 2 E \partial_k^2\mathcal{F}=((c_k\ell+2\mathcal{B}_k)^2-2\mathcal{B}_k^2)E\ge-2\mathcal{B}_k^2E ∂ k 2 F = (( c k ℓ + 2 B k ) 2 − 2 B k 2 ) E ≥ − 2 B k 2 E w k 2 / 8 w_k^2/8 w k 2 /8
F − I k F ≤ 2 B k 2 ∫ G E ≤ w k 2 4 B k 2 ∑ i ω i ( k ) E ( b i ) \mathcal{F}-I_k\mathcal{F}\le2\mathcal{B}_k^2\int GE\le\frac{w_k^2}{4}\mathcal{B}_k^2\sum_i\omega_i^{(k)}E(b_i) F − I k F ≤ 2 B k 2 ∫ GE ≤ 4 w k 2 B k 2 i ∑ ω i ( k ) E ( b i ) by the left inequality of (12), with weights ω i ( k ) ≥ 0 \omega_i^{(k)}\ge0 ω i ( k ) ≥ 0 U U U I j I_j I j U U U Z ( b ) Z(b) Z ( b )
The cover and the theorem
Fix a box Q Q Q C C C κ \kappa κ Var ( X h ) ≤ ν b [ ( X h − κ ) 2 ] \operatorname{Var}(X_h)\le\nu_b[(X_h-\kappa)^2] Var ( X h ) ≤ ν b [( X h − κ ) 2 ] κ = ⟨ κ ( b ) , h ⟩ \kappa=\langle\kappa(b),h\rangle κ = ⟨ κ ( b ) , h ⟩ κ ( b ) \kappa(b) κ ( b ) b b b X h − κ = A + ∑ k ( b k − β k ) B k X_h-\kappa=\mathcal{A}+\sum_k(b_k-\beta_k)\mathcal{B}_k X h − κ = A + ∑ k ( b k − β k ) B k β \beta β Q Q Q B k = ⟨ ∇ φ F k , ∇ q h ⟩ − ⟨ ∂ k κ , h ⟩ \mathcal{B}_k=\langle\nabla\varphi_{F_k},\nabla q_h\rangle-\langle\partial_k\kappa,h\rangle B k = ⟨ ∇ φ F k , ∇ q h ⟩ − ⟨ ∂ k κ , h ⟩ b ∈ Q ∩ R r b\in Q\cap\mathcal{R}_r b ∈ Q ∩ R r
sup h L h ( b ) ≤ e ς / 8 max i λ A ( b i ) + e ς / 8 γ ( r ) ( max i λ V ( b i ) + e N min i Z ( b i ) ) . (13) \sup_h L_h(b)\le\sqrt{e^{\varsigma/8}\max_i\lambda_{\mathcal{A}}(b_i)}
+\sqrt{\frac{e^{\varsigma/8}}{\gamma(r)}\left(\max_i\lambda_V(b_i)+\frac{e_N}{\min_i Z(b_i)}\right)}. \tag*{(13)} h sup L h ( b ) ≤ e ς /8 i max λ A ( b i ) + γ ( r ) e ς /8 ( i max λ V ( b i ) + min i Z ( b i ) e N ) . (13) where e N = ∑ k w k 2 4 max i Z ( b i ) λ k ( b i ) e_N=\sum_k\frac{w_k^2}{4}\max_i Z(b_i)\lambda_k(b_i) e N = ∑ k 4 w k 2 max i Z ( b i ) λ k ( b i ) λ A \lambda_{\mathcal{A}} λ A λ V \lambda_V λ V λ k \lambda_k λ k h ↦ ν b i ∣ ∇ q h ∣ 2 h\mapsto\nu_{b_i}|\nabla q_h|^2 h ↦ ν b i ∣∇ q h ∣ 2 h ↦ ν b i [ ( X h − κ ) 2 ] h\mapsto\nu_{b_i}[(X_h-\kappa)^2] h ↦ ν b i [( X h − κ ) 2 ] b = b i b=b_i b = b i h ↦ ν b i [ B k 2 ] h\mapsto\nu_{b_i}[\mathcal{B}_k^2] h ↦ ν b i [ B k 2 ] R 18 \mathbb{R}^{18} R 18 ν b i \nu_{b_i} ν b i T ^\mathsf{T} T t I tI t I [43 , 48 ] .
The matrix C C C b = β b=\beta b = β u = q h u=q_h u = q h C C C θ > 0 \theta>0 θ > 0 ∣ ∇ q ∣ 2 + θ Var ( X ) / γ \lvert\nabla q\rvert^2+\theta\operatorname{Var}(X)/\gamma ∣ ∇ q ∣ 2 + θ Var ( X ) / γ h h h θ \theta θ C C C 2 − 30 2^{-30} 2 − 30
A box is discarded when it contains no point with a 1 ≥ a 2 ≥ ∣ z 3 ∣ a_1\ge a_2\ge|z_3| a 1 ≥ a 2 ≥ ∣ z 3 ∣ ℓ ( b ) = Re Tr ( c ∗ b ) \ell(b)=\operatorname{Re}\operatorname{Tr}(c^*b) ℓ ( b ) = Re Tr ( c ∗ b ) min Q ℓ > r max T ℓ \min_Q\ell>r\max_T\ell min Q ℓ > r max T ℓ r conv T r\operatorname{conv}T r conv T R r \mathcal{R}_r R r
Proposition 34. 2 r Λ ( r ) ≤ 0.999887 2r\Lambda(r)\le0.999887 2 r Λ ( r ) ≤ 0.999887 r = 3 2 r=\frac{3}{2} r = 2 3
Proof. The box [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^2\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] r / 8 r/8 r /8 B Q B_Q B Q γ ( r ) = 187 128 \gamma(r)=\frac{187}{128} γ ( r ) = 128 187 2 r B Q < 1 2rB_Q<1 2 r B Q < 1 R r \mathcal{R}_r R r [ 1.3125 , 1.5 ] × [ 0.9375 , 1.125 ] × [ 0.5625 , 0.75 ] × [ 0 , 0.1875 ] [1.3125,1.5]\times[0.9375,1.125]\times[0.5625,0.75]\times[0,0.1875] [ 1.3125 , 1.5 ] × [ 0.9375 , 1.125 ] × [ 0.5625 , 0.75 ] × [ 0 , 0.1875 ] Figure 6 shows the cover, and Appendix E the error analysis.
Figure 6. The 2341 certified boxes at r = 3 2 r = \frac{3}{2} r = 2 3 ( a 1 , a 2 ) (a_1,a_2) ( a 1 , a 2 ) diag ( a 1 , a 2 , x 3 + i y 3 ) \operatorname{diag}(a_1,a_2,x_3+\mathrm{i}y_3) diag ( a 1 , a 2 , x 3 + i y 3 ) 2 r sup h L h 2r\sup_h L_h 2 r sup h L h r / 8 r/8 r /8 ∣ x 3 + i y 3 ∣ \lvert x_3+\mathrm{i}y_3\rvert ∣ x 3 + i y 3 ∣ x 3 x_3 x 3 R r \mathcal{R}_r R r
The same cover at a larger r r r Λ ( r ) \Lambda(r) Λ ( r ) 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r )
Proposition 35. 2 r Λ ( r ) ≤ 1.029967 2r\Lambda(r)\le1.029967 2 r Λ ( r ) ≤ 1.029967 r = 38 25 r=\frac{38}{25} r = 25 38
Proof. As in the proof of Proposition 34, with γ ( r ) = 4643 3200 \gamma(r)=\frac{4643}{3200} γ ( r ) = 3200 4643 r / 8 r/8 r /8 R r \mathcal{R}_r R r
Proof of Theorem 6. Proposition 34 gives
Λ ( 3 2 ) ≤ 0.999887 3 < 0.333296. (14) \Lambda\left(\frac{3}{2}\right)\le\frac{0.999887}{3}<0.333296. \tag*{(14)} Λ ( 2 3 ) ≤ 3 0.999887 < 0.333296. (14) For β W ≤ 3 4 \beta_W \le\frac{3}{4} β W ≤ 4 3 2 β W ≤ 3 2 2\beta_W \le\frac{3}{2} 2 β W ≤ 2 3 (8) with the monotonicity of Λ \Lambda Λ
α ≤ 4 β W Λ ( 3 2 ) < 1.333183 β W ≤ 3 4 ⋅ 1.333183 < 1. \alpha\le4\beta_W\Lambda\left(\frac{3}{2}\right)<1.333183\beta_W\le\frac{3}{4}\cdot1.333183<1. α ≤ 4 β W Λ ( 2 3 ) < 1.333183 β W ≤ 4 3 ⋅ 1.333183 < 1. Proposition 11 applies with the Hilbert–Schmidt metric.
SU(3) WITH PLAQUETTE BLOCKS
In this section G = S U ( 3 ) G=\mathrm{SU}(3) G = SU ( 3 ) β ≥ 0 \beta\ge0 β ≥ 0 β W = 9 β \beta_W=9\beta β W = 9 β N β = β W / 3 N\beta=\beta_W/3 N β = β W /3 d d d K = conv S U ( 3 ) K=\operatorname{conv}\mathrm{SU}(3) K = conv SU ( 3 ) ν c ( d U ) ∝ e Re Tr ( U c ) d U \nu_c(dU)\propto e^{\operatorname{Re}\operatorname{Tr}(Uc)}dU ν c ( d U ) ∝ e Re Tr ( U c ) d U ϕ h ( U ) = Re Tr ( U h ) \phi_h(U)=\operatorname{Re}\operatorname{Tr}(Uh) ϕ h ( U ) = Re Tr ( U h ) Λ ( r ) \Lambda(r) Λ ( r ) (8) . Put Z ( c ) = ∫ e Re Tr ( U c ) d U Z(c)=\int e^{\operatorname{Re}\operatorname{Tr}(Uc)}dU Z ( c ) = ∫ e Re Tr ( U c ) d U g = log Z g=\log Z g = log Z R 18 \mathbb{R}^{18} R 18 R 18 \mathbb{R}^{18} R 18 ⟨ x , y ⟩ = Re Tr ( x y ∗ ) \langle x,y\rangle=\operatorname{Re}\operatorname{Tr}(xy^*) ⟨ x , y ⟩ = Re Tr ( x y ∗ ) g g g c c c m ( c ) ∗ m(c)^* m ( c ) ∗ m ( c ) = ν c ( U ) ∈ K m(c)=\nu_c(U)\in K m ( c ) = ν c ( U ) ∈ K Hess g ( c ) ( h , h ) = Var ν c ( ϕ h ) \operatorname{Hess}g(c)(h,h)=\operatorname{Var}_{\nu_c}(\phi_h) Hess g ( c ) ( h , h ) = Var ν c ( ϕ h ) Z ( V c W ) = Z ( c ) Z(VcW)=Z(c) Z ( V c W ) = Z ( c ) V , W ∈ S U ( 3 ) V,W\in\mathrm{SU}(3) V , W ∈ SU ( 3 ) g g g
The blocks are those of Lemma 12, which concerns the line field only and holds for every group. For a block plaquette with active links e 1 , e 2 , e 3 e_1,e_2,e_3 e 1 , e 2 , e 3 Re Tr Q p = Re Tr ( y 1 y 2 y 3 ) \operatorname{Re}\operatorname{Tr}Q_p=\operatorname{Re}\operatorname{Tr}(y_1y_2y_3) Re Tr Q p = Re Tr ( y 1 y 2 y 3 ) y i = Q e i y_i=Q_{e_i} y i = Q e i Q e i − 1 Q_{e_i}^{-1} Q e i − 1
μ b ( d y ) ∝ exp ( ∑ i Re Tr ( y i b i ) + N β Re Tr ( y 1 y 2 y 3 ) ) d y , (15) \mu_b(dy)\propto\exp\left(\sum_i\operatorname{Re}\operatorname{Tr}(y_ib_i)+N\beta\operatorname{Re}\operatorname{Tr}(y_1y_2y_3)\right)dy, \tag*{(15)} μ b ( d y ) ∝ exp ( i ∑ Re Tr ( y i b i ) + N β Re Tr ( y 1 y 2 y 3 ) ) d y , (15) where b i b_i b i N β N\beta N β y i y_i y i b i ∈ 5 N β K b_i\in5N\beta K b i ∈ 5 N β K ∥ b i ∥ o p ≤ 5 N β \lVert b_i\rVert_{\mathrm{op}}\le5N\beta ∥ b i ∥ op ≤ 5 N β S U ( 3 ) 3 \mathrm{SU}(3)^3 SU ( 3 ) 3 d ( y , y ′ ) = ∑ i ∥ y i − y i ′ ∥ H S d(y,y')=\sum_i\lVert y_i-y_i'\rVert_{\mathrm{HS}} d ( y , y ′ ) = ∑ i ∥ y i − y i ′ ∥ HS
The sensitivity of one block
Let r = 2 β W = 6 N β r=2\beta_W=6N\beta r = 2 β W = 6 N β λ ∗ \lambda_* λ ∗ τ \tau τ Hess g \operatorname{Hess}g Hess g r K rK r K M M M r K rK r K ∣ D 3 g ( c ) ( u , v , v ) ∣ \lvert D^3g(c)(u,v,v)\rvert ∣ D 3 g ( c ) ( u , v , v )∣ u , v u,v u , v γ = γ ( r ) \gamma=\gamma(r) γ = γ ( r )
w = 3 2 ( N β ) 2 ( τ − λ ∗ ) , s = Λ ( r ) + ( e w − 1 ) e w τ γ , w=\frac{3}{2}(N\beta)^2(\tau-\lambda_*),\qquad
s=\Lambda(r)+(e^w-1)e^w\sqrt{\frac{\tau}{\gamma}}, w = 2 3 ( N β ) 2 ( τ − λ ∗ ) , s = Λ ( r ) + ( e w − 1 ) e w γ τ , C = 5 ( N β ) 2 ( τ s + 3 N β M e w γ ) , C=5(N\beta)^2\left(\tau s+\frac{\sqrt{3}N\beta M e^w}{\gamma}\right), C = 5 ( N β ) 2 ( τ s + γ 3 N β M e w ) , K 3 = ( 1 + N β Λ ( r ) ) s 1 − C , α 3 = 10 N β K 3 , (16) K_3=\frac{(1+N\beta\Lambda(r))s}{1-C},\qquad
\alpha_3=10N\beta K_3, \tag*{(16)} K 3 = 1 − C ( 1 + N β Λ ( r )) s , α 3 = 10 N β K 3 , (16) assuming C < 1 C<1 C < 1
Lemma 36. Let a , c 0 ∈ 5 N β K a,c_0\in5N\beta K a , c 0 ∈ 5 N β K
κ ( d x ) ∝ exp ( Re Tr ( x a ) + g ( c 0 + N β x ) ) d x on S U ( 3 ) . \kappa(dx)\propto\exp\left(\operatorname{Re}\operatorname{Tr}(xa)+g(c_0+N\beta x)\right)dx\quad\text{on }\mathrm{SU}(3). κ ( d x ) ∝ exp ( Re Tr ( x a ) + g ( c 0 + N β x ) ) d x on SU ( 3 ) . For every 1 1 1 f f f h ∈ C 3 × 3 h\in\mathbb{C}^{3\times3} h ∈ C 3 × 3
(i) Cov κ ( f , ϕ h ) ≤ s ∥ h ∥ H S \operatorname{Cov}_{\kappa}(f,\phi_h)\le s\lVert h\rVert_{\mathrm{HS}} Cov κ ( f , ϕ h ) ≤ s ∥ h ∥ HS
(ii) if c 0 c_0 c 0 k k k 5 N β K 5N\beta K 5 N β K
∣ d d t κ ( f ) ∣ ≤ N β ( τ s + 3 N β M e w γ ) ∥ k ∥ H S . \left\lvert\frac{d}{dt}\kappa(f)\right\rvert\le N\beta\left(\tau s+\frac{\sqrt{3}N\beta M e^w}{\gamma}\right)\lVert k\rVert_{\mathrm{HS}}. d t d κ ( f ) ≤ N β ( τ s + γ 3 N β M e w ) ∥ k ∥ HS . Proof. (i) The segment from c 0 c_0 c 0 c 0 + N β x c_0+N\beta x c 0 + N β x 6 N β K = r K 6N\beta K=rK 6 N β K = r K
g ( c 0 + N β x ) = g ( c 0 ) + N β Re Tr ( x m ( c 0 ) ) + R ( x ) g(c_0+N\beta x)=g(c_0)+N\beta\operatorname{Re}\operatorname{Tr}(xm(c_0))+R(x) g ( c 0 + N β x ) = g ( c 0 ) + N β Re Tr ( x m ( c 0 )) + R ( x ) with R ( x ) = 1 2 ( N β ) 2 Hess g ( ξ ) ( x , x ) R(x)=\frac{1}{2}(N\beta)^2\operatorname{Hess}g(\xi)(x,x) R ( x ) = 2 1 ( N β ) 2 Hess g ( ξ ) ( x , x ) ∥ x ∥ H S 2 = 3 \lVert x\rVert_{\mathrm{HS}}^2=3 ∥ x ∥ HS 2 = 3 R ∈ [ 3 2 ( N β ) 2 λ ∗ , 3 2 ( N β ) 2 τ ] R\in[\frac{3}{2}(N\beta)^2\lambda_*,\frac{3}{2}(N\beta)^2\tau] R ∈ [ 2 3 ( N β ) 2 λ ∗ , 2 3 ( N β ) 2 τ ] osc R ≤ w \operatorname{osc}R\le w osc R ≤ w κ \kappa κ e R ν / ν ( e R ) e^R\nu/\nu(e^R) e R ν / ν ( e R ) ν = ν c ′ \nu=\nu_{c'} ν = ν c ′ c ′ = a + N β m ( c 0 ) ∈ r K c'=a+N\beta m(c_0)\in rK c ′ = a + N β m ( c 0 ) ∈ r K Λ \Lambda Λ Cov ν ( f , ϕ h ) ≤ Λ ( r ) ∥ h ∥ \operatorname{Cov}_{\nu}(f,\phi_h)\le\Lambda(r)\lVert h\rVert Cov ν ( f , ϕ h ) ≤ Λ ( r ) ∥ h ∥ f f f 1 1 1 d d d sd ν f ≤ γ − 1 / 2 \operatorname{sd}_{\nu}f\le\gamma^{-1/2} sd ν f ≤ γ − 1/2 sd ν ϕ h ≤ τ ∥ h ∥ \operatorname{sd}_{\nu}\phi_h\le\sqrt{\tau}\lVert h\rVert sd ν ϕ h ≤ τ ∥ h ∥
(ii) The derivative is Cov κ ( f , ⟨ ∇ g ( c 0 + N β x ) , k ⟩ ) \operatorname{Cov}_{\kappa}(f,\langle\nabla g(c_0+N\beta x),k\rangle) Cov κ ( f , ⟨ ∇ g ( c 0 + N β x ) , k ⟩) ∇ g ( c 0 + N β u ) = ∇ g ( c 0 ) + N β Hess g ( c 0 ) u + N ( u ) \nabla g(c_0+N\beta u)=\nabla g(c_0)+N\beta\operatorname{Hess}g(c_0)u+N(u) ∇ g ( c 0 + N β u ) = ∇ g ( c 0 ) + N β Hess g ( c 0 ) u + N ( u ) ϕ h \phi_h ϕ h ∥ h ∥ ≤ N β τ ∥ k ∥ \lVert h\rVert\le N\beta\tau\lVert k\rVert ∥ h ∥ ≤ N β τ ∥ k ∥ N β τ s ∥ k ∥ N\beta\tau s\lVert k\rVert N β τ s ∥ k ∥ c 0 c_0 c 0 N β M ∥ u ∥ ≤ 3 N β M N\beta M\lVert u\rVert\le\sqrt{3}N\beta M N β M ∥ u ∥ ≤ 3 N β M x ↦ ⟨ N ( x ) , k ⟩ x\mapsto\langle N(x),k\rangle x ↦ ⟨ N ( x ) , k ⟩ 3 ( N β ) 2 M ∥ k ∥ 3(N\beta)^2M\lVert k\rVert 3 ( N β ) 2 M ∥ k ∥ d d d κ \kappa κ ν \nu ν w w w e − w γ e^{-w}\gamma e − w γ [44 ] , and the covariance of two Lipschitz functions under κ \kappa κ e w / γ e^w/\gamma e w / γ
The constants and the proof of Theorem 7
Lemma 37. Let C < 1 C<1 C < 1 b , b ′ b,b' b , b ′ 5 N β K 5N\beta K 5 N β K b i b_i b i W 1 ( μ b , μ b ′ ) ≤ K 3 ∥ b i − b i ′ ∥ H S W_1(\mu_b,\mu_{b'})\le K_3\lVert b_i-b_i'\rVert_{\mathrm{HS}} W 1 ( μ b , μ b ′ ) ≤ K 3 ∥ b i − b i ′ ∥ HS d d d
Proof. By the cyclic symmetry of Re Tr ( y 1 y 2 y 3 ) \operatorname{Re}\operatorname{Tr}(y_1y_2y_3) Re Tr ( y 1 y 2 y 3 ) i = 1 i=1 i = 1 y 1 , y 3 y_1,y_3 y 1 , y 3 y 2 y_2 y 2 ν c \nu_c ν c c = b 2 + N β y 3 y 1 ∈ r K c=b_2+N\beta y_3y_1\in rK c = b 2 + N β y 3 y 1 ∈ r K y 1 y_1 y 1 η \eta η d d d c c c N β η N\beta\eta N β η ν c \nu_c ν c N β Λ ( r ) η N\beta\Lambda(r)\eta N β Λ ( r ) η W 1 W_1 W 1 (8) ; so F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] F(y_1,y_3)=\mathbb{E}[f\mid y_1,y_3] F ( y 1 , y 3 ) = E [ f ∣ y 1 , y 3 ] ( 1 + N β Λ ( r ) ) (1+N\beta\Lambda(r)) ( 1 + N β Λ ( r )) ( y 1 , y 3 ) (y_1,y_3) ( y 1 , y 3 ) e Re Tr ( y 1 b 1 ) + Re Tr ( y 3 b 3 ) Z ( b 2 + N β y 3 y 1 ) e^{\operatorname{Re}\operatorname{Tr}(y_1b_1)+\operatorname{Re}\operatorname{Tr}(y_3b_3)}Z(b_2+N\beta y_3y_1) e Re Tr ( y 1 b 1 ) + Re Tr ( y 3 b 3 ) Z ( b 2 + N β y 3 y 1 ) Z ( b 2 + N β y 3 y 1 ) = Z ( y 3 − 1 b 2 + N β y 1 ) = Z ( b 2 y 1 − 1 + N β y 3 ) Z(b_2+N\beta y_3y_1)=Z(y_3^{-1}b_2+N\beta y_1)=Z(b_2y_1^{-1}+N\beta y_3) Z ( b 2 + N β y 3 y 1 ) = Z ( y 3 − 1 b 2 + N β y 1 ) = Z ( b 2 y 1 − 1 + N β y 3 ) c 0 = y 3 − 1 b 2 c_0=y_3^{-1}b_2 c 0 = y 3 − 1 b 2 c 0 = b 2 y 1 − 1 c_0=b_2y_1^{-1} c 0 = b 2 y 1 − 1 y 3 y_3 y 3 y 1 y_1 y 1 y 3 − 1 b 2 y_3^{-1}b_2 y 3 − 1 b 2 5 N β K 5N\beta K 5 N β K ∥ b 2 ∥ o p ∥ y 3 − y 3 ′ ∥ ≤ 5 N β ∥ y 3 − y 3 ′ ∥ \lVert b_2\rVert_{\mathrm{op}}\lVert y_3-y_3'\rVert\le5N\beta\lVert y_3-y_3'\rVert ∥ b 2 ∥ op ∥ y 3 − y 3 ′ ∥ ≤ 5 N β ∥ y 3 − y 3 ′ ∥ d 1 ≤ s ε + C d 3 d_1\le s\varepsilon+Cd_3 d 1 ≤ s ε + C d 3 d 3 ≤ C d 1 d_3\le Cd_1 d 3 ≤ C d 1 ∣ μ b ( f ) − μ b ′ ( f ) ∣ ≤ ( 1 + N β Λ ( r ) ) ( d 1 + d 3 ) ≤ K 3 ε \lvert\mu_b(f)-\mu_{b'}(f)\rvert\le(1+N\beta\Lambda(r))(d_1+d_3)\le K_3\varepsilon ∣ μ b ( f ) − μ b ′ ( f )∣ ≤ ( 1 + N β Λ ( r )) ( d 1 + d 3 ) ≤ K 3 ε
Lemma 38. At r = 3 2 r=\frac{3}{2} r = 2 3 Hess g \operatorname{Hess}g Hess g r K rK r K [ 0.056015 , 0.247316 ] [0.056015,0.247316] [ 0.056015 , 0.247316 ] ∣ D 3 g ( c ) ( u , v , v ) ∣ ≤ 0.182126 \lvert D^3g(c)(u,v,v)\rvert\le0.182126 ∣ D 3 g ( c ) ( u , v , v )∣ ≤ 0.182126 c ∈ r K c\in rK c ∈ r K u , v u,v u , v r = 38 25 r=\frac{38}{25} r = 25 38 [ 0.054978 , 0.248280 ] [0.054978,0.248280] [ 0.054978 , 0.248280 ] 0.183546 0.183546 0.183546
Proof. For V , W ∈ S U ( 3 ) V,W\in\mathrm{SU}(3) V , W ∈ SU ( 3 ) g ( V c W ) = g ( c ) g(VcW)=g(c) g ( V c W ) = g ( c ) x ↦ V x W x\mapsto VxW x ↦ V x W Hess g \operatorname{Hess}g Hess g D 3 g D^3g D 3 g r K rK r K g g g ϕ \phi ϕ ν c \nu_c ν c [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] [0,r]^2\times[-r,r]\times[0,r] [ 0 , r ] 2 × [ − r , r ] × [ 0 , r ] r / 8 r/8 r /8 r = 38 25 r=\frac{38}{25} r = 25 38 Q Q Q ϕ \phi ϕ b i b_i b i b 0 b_0 b 0 k = m ( b 0 ) k=m(b_0) k = m ( b 0 ) ϕ \phi ϕ Upper bound. For b ∈ Q b \in Q b ∈ Q h h h Var ν b ( ϕ h ) ≤ ν b [ ( ϕ h − ⟨ k , h ⟩ ) 2 ] ≤ e ε / 8 max i ν b i [ ( ϕ h − ⟨ k , h ⟩ ) 2 ] \operatorname{Var}_{\nu_b}(\phi_h) \le\nu_b[(\phi_h-\langle k,h\rangle)^2] \le e^{\varepsilon/8}\max_i\nu_{b_i}[(\phi_h-\langle k,h\rangle)^2] Var ν b ( ϕ h ) ≤ ν b [( ϕ h − ⟨ k , h ⟩ ) 2 ] ≤ e ε /8 max i ν b i [( ϕ h − ⟨ k , h ⟩ ) 2 ]
Lower bound. Integrating the right inequality of (12) against f ≥ 0 f \ge0 f ≥ 0 Z ( b ) ≤ ∑ i ω i Z ( b i ) Z(b) \le\sum_i\omega_i Z(b_i) Z ( b ) ≤ ∑ i ω i Z ( b i ) ν b ( f ) ≥ e − ε / 8 min i ν b i ( f ) \nu_b(f) \ge e^{-\varepsilon/8}\min_i\nu_{b_i}(f) ν b ( f ) ≥ e − ε /8 min i ν b i ( f ) f = ( ϕ h − ν b ϕ h ) 2 f=(\phi_h-\nu_b\phi_h)^2 f = ( ϕ h − ν b ϕ h ) 2 Var ν b ( ϕ h ) ≥ e − ε / 8 min i Var ν b i ( ϕ h ) \operatorname{Var}_{\nu_b}(\phi_h) \ge e^{-\varepsilon/8}\min_i\operatorname{Var}_{\nu_{b_i}}(\phi_h) Var ν b ( ϕ h ) ≥ e − ε /8 min i Var ν b i ( ϕ h )
Third cumulant. Put δ = m ( b ) − k \delta=m(b)-k δ = m ( b ) − k ∥ δ ∥ ≤ τ ρ + e \lVert\delta\rVert\le\tau\rho+e ∥ δ ∥ ≤ τ ρ + e ρ \rho ρ b 0 b_0 b 0 Q Q Q e e e k k k
D 3 g ( b ) ( u , v , v ) = T b ( u , v , v ) − δ u Var ( ϕ v ) − 2 δ v Cov ( ϕ u , ϕ v ) − δ u δ v 2 , D^3g(b)(u,v,v)=T_b(u,v,v)-\delta_u\operatorname{Var}(\phi_v)-2\delta_v\operatorname{Cov}(\phi_u,\phi_v)-\delta_u\delta_v^2, D 3 g ( b ) ( u , v , v ) = T b ( u , v , v ) − δ u Var ( ϕ v ) − 2 δ v Cov ( ϕ u , ϕ v ) − δ u δ v 2 , with T b = ν b [ ( ϕ − k ) ⊗ 3 ] T_b=\nu_b[(\phi-k)^{\otimes3}] T b = ν b [( ϕ − k ) ⊗ 3 ] δ u = ⟨ δ , u ⟩ \delta_u=\langle\delta,u\rangle δ u = ⟨ δ , u ⟩ ν b \nu_b ν b G G G G = G + − G − G=G^+-G^- G = G + − G − G ± G^\pm G ± ∣ ν b ( G ) ∣ ≤ ( max i ∣ ν b i ( G ) ∣ + ε max i ν b i ∣ G ∣ ) / ( 1 − ε ) \lvert\nu_b(G)\rvert\le(\max_i\lvert\nu_{b_i}(G)\rvert+\varepsilon\max_i\nu_{b_i}\lvert G\rvert)/(1-\varepsilon) ∣ ν b ( G )∣ ≤ ( max i ∣ ν b i ( G )∣ + ε max i ν b i ∣ G ∣) / ( 1 − ε ) ε = 1 − e − ε / 8 \varepsilon=1-e^{-\varepsilon/8} ε = 1 − e − ε /8 G = ( ϕ u − k u ) ( ϕ v − k v ) 2 G=(\phi_u-k_u)(\phi_v-k_v)^2 G = ( ϕ u − k u ) ( ϕ v − k v ) 2 ∣ ν b i ( G ) ∣ ≤ ∥ T b i ∥ \lvert\nu_{b_i}(G)\rvert\le\lVert T_{b_i}\rVert ∣ ν b i ( G )∣ ≤ ∥ T b i ∥ 18 × 324 18\times324 18 × 324 T b i T_{b_i} T b i ν b i ∣ G ∣ ≤ 2 3 ν b i [ ( ϕ v − k v ) 2 ] \nu_{b_i}\lvert G\rvert\le2\sqrt{3}\nu_{b_i}[(\phi_v-k_v)^2] ν b i ∣ G ∣ ≤ 2 3 ν b i [( ϕ v − k v ) 2 ] ∥ U − k ∥ H S ≤ 2 3 \lVert U-k\rVert_{\mathrm{HS}}\le2\sqrt{3} ∥ U − k ∥ HS ≤ 2 3
Proof of Theorem 7. Let first β W ≤ 3 4 \beta_W\le\frac{3}{4} β W ≤ 4 3 r = 2 β W ≤ 3 2 r=2\beta_W\le\frac{3}{2} r = 2 β W ≤ 2 3 3 2 K \frac{3}{2}K 2 3 K 3 2 \frac{3}{2} 2 3 Λ ( 3 2 ) ≤ 0.999887 3 \Lambda(\frac{3}{2})\le\frac{0.999887}{3} Λ ( 2 3 ) ≤ 3 0.999887 γ ( 3 2 ) = 187 128 \gamma(\frac{3}{2})=\frac{187}{128} γ ( 2 3 ) = 128 187 (16) is increasing in β W \beta_W β W 3 4 < β W ≤ 19 25 \frac{3}{4}<\beta_W\le\frac{19}{25} 4 3 < β W ≤ 25 19 38 25 K \frac{38}{25}K 25 38 K 38 25 \frac{38}{25} 25 38 Λ ( 38 25 ) ≤ 1.029967 3.04 \Lambda(\frac{38}{25})\le\frac{1.029967}{3.04} Λ ( 25 38 ) ≤ 3.04 1.029967 γ ( 38 25 ) = 46433200 ( \gamma(\frac{38}{25})=\frac{46433200}{\mathstrut\ } γ ( 25 38 ) = ( 46433200 j j j η \eta η η \eta η b i b_i b i N β η N\beta\eta N β η ∑ B ⊂ B j C B ≤ 10 N β K 3 = α 3 ( β W ) \sum_{B\subset B_j}C_B\le10N\beta K_3=\alpha_3(\beta_W) ∑ B ⊂ B j C B ≤ 10 N β K 3 = α 3 ( β W ) β W = 3 4 \beta_W=\frac{3}{4} β W = 4 3 w ≤ 0.017935 w\le0.017935 w ≤ 0.017935 s ≤ 0.34088 s\le0.34088 s ≤ 0.34088 C ≤ 0.043520 C\le0.043520 C ≤ 0.043520 K 3 ≤ 0.38609 K_3\le0.38609 K 3 ≤ 0.38609 α 3 ≤ 0.96521 \alpha_3\le0.96521 α 3 ≤ 0.96521 β W = 0.72 \beta_W=0.72 β W = 0.72 α 3 ≤ 0.91815 \alpha_3\le0.91815 α 3 ≤ 0.91815 β W = 0.76 \beta_W=0.76 β W = 0.76 w ≤ 0.018609 w\le0.018609 w ≤ 0.018609 s ≤ 0.34673 s\le0.34673 s ≤ 0.34673 C ≤ 0.045770 C\le0.045770 C ≤ 0.045770 K 3 ≤ 0.39454 K_3\le0.39454 K 3 ≤ 0.39454 α 3 ≤ 0.99950 \alpha_3\le0.99950 α 3 ≤ 0.99950
Remark 39. Theorem 6 ends where $2r\Lambda(r)$ reaches one. The blocks remove the factor 12 10 \frac{12}{10} 10 12 β W = 3 4 \beta_W=\frac{3}{4} β W = 4 3 1.1 ⋅ 10 − 4 1.1\cdot10^{-4} 1.1 ⋅ 1 0 − 4 β W ≥ 0.715 \beta_W\ge0.715 β W ≥ 0.715 10 12 \frac{10}{12} 12 10 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) 2 r Λ ( r ) ≤ 1.03 2r\Lambda(r)\le1.03 2 r Λ ( r ) ≤ 1.03 r = 1.52 r=1.52 r = 1.52 β W = 0.76 \beta_W=0.76 β W = 0.76 R r \mathcal{R}_r R r
Where the methods stop
SU(2). Theorem 2 closes the single-link route in every forest gauge at β W = 1 3 \beta_W=\frac{1}{3} β W = 3 1 [31 ] , pass it: plaquette blocks reach β W = 0.345 \beta_{\mathrm W}=0.345 β W = 0.345 Theorem 3 ), and the sharper floating-point bound of Remark 18 stops near 0.357. Larger blocks are the next step.
Curvature. Remark 26 locates the end of the curvature method in this gauge between ∣ β ∣ = 1 29 |\beta|=\frac{1}{29} ∣ β ∣ = 29 1 1 26.25 \frac{1}{26.25} 26.25 1
SU(3). The route of Sec. VII bounds the H − 1 ( ν b ) H^{-1}(\nu_b) H − 1 ( ν b ) φ h − ν b φ h \varphi_h-\nu_b\varphi_h φ h − ν b φ h L h ( b ) L_h(b) L h ( b ) r K rK r K r K rK r K 2 r 2r 2 r r = 1.25 r=1.25 r = 1.25 r = 1.4 r=1.4 r = 1.4 r = 1.5 r=1.5 r = 1.5 r = 1.55 r=1.55 r = 1.55 β W ≈ 0.77 \beta_{\mathrm W}\approx0.77 β W ≈ 0.77 u = s φ h u=s\varphi_h u = s φ h R r \mathcal{R}_r R r diag ( r , ρ , ρ ) \operatorname{diag}(r,\rho,\rho) diag ( r , ρ , ρ ) ρ \rho ρ 0.75 r 0.75r 0.75 r 0.8 r 0.8r 0.8 r 2 r 2r 2 r r = 1.36 r=1.36 r = 1.36 r = 1.40 r=1.40 r = 1.40 r = 1.44 r=1.44 r = 1.44 β W ≈ 0.715 \beta_{\mathrm W}\approx0.715 β W ≈ 0.715 u = q h u=q_h u = q h r = 1.44 r=1.44 r = 1.44 r = 1.50 r=1.50 r = 1.50 r = 1.54 r=1.54 r = 1.54 Figure 7 ), close to the Galerkin values. Lemma 29 therefore costs little at these couplings, and most of the distance between the floating-point value at r = 1.5 r=1.5 r = 1.5 β W = 3 4 \beta_{\mathrm W}=\frac{3}{4} β W = 4 3 Theorem 7 use Λ ( 2 β W ) \Lambda(2\beta_{\mathrm W}) Λ ( 2 β W )
Figure 7. The floating-point value of 2 r sup h 2r\sup h 2 r sup h b = diag ( r , ρ , ρ ) b=\operatorname{diag}(r,\rho,\rho) b = diag ( r , ρ , ρ ) ρ / r \rho/r ρ / r r = 1.44 r=1.44 r = 1.44 1.50 1.50 1.50 1.54 1.54 1.54 u = s φ h u=s\varphi_h u = s φ h s s s u = q h u=q_h u = q h C C C r = 1.44 r=1.44 r = 1.44 r = 1.50 r=1.50 r = 1.50 R r \mathcal{R}_r R r r = 1.54 r=1.54 r = 1.54
Other groups. For groups other than SU(2) and SU(3) only the curvature route is carried out here (Theorem 5 ), and it stops where the bound 29 2 \frac{29}{2} 2 29 G G G N ∣ β ∣ = 29 2 κ G N|\beta|=\frac{29}{2}\kappa_G N ∣ β ∣ = 2 29 κ G
The four open problems. Our earlier paper [14 ] listed four problems between the strong-coupling results and the Clay problem [47 ] : (1) uniqueness and a gap at intermediate fixed coupling; (2) the crossover from strong coupling to Bałaban’s weak-coupling regime; (3) the continuum limit with the Osterwalder–Schrader axioms; and (4) a gap in physical units. The present paper moves the boundary of (1): for SU(2) from β W = 2 9 \beta_{\mathrm W}=\frac{2}{9} β W = 9 2 9 32 \frac{9}{32} 32 9 3 4 \frac{3}{4} 4 3 β W ≈ 2.2 \beta_{\mathrm W}\approx2.2 β W ≈ 2.2 β W ≈ 6 \beta_{\mathrm W}\approx6 β W ≈ 6 [15 ] none of them can by itself produce a continuum mass gap; that requires control of the gap in physical units along the renormalisation group flow [5 , 17 ] .13
The continuum problem
Two general facts limit what can be imported into problems (3) and (4) above from constructions of continuum measures. Both concern a random distribution F F F R d \mathbb{R}^d R d
E F ( χ ) ≤ e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) (17) \mathbb{E}F(\chi)\leq e^{C(\|\chi\|_1+\|\chi\|_2^2)} \tag*{(17)} E F ( χ ) ≤ e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) (17) for all real χ ∈ S ( R d ) \chi\in\mathcal{S}(\mathbb{R}^d) χ ∈ S ( R d ) p = 2 p=2 p = 2 [39 ] , Sect. 6.1, (6.1.5). It is the hypothesis under which reflection positivity across coordinate and diagonal hyperplanes is shown in [73 ] to give rotation invariance of the Schwinger functions, a step proposed there for the continuum limit of lattice gauge theories.
Proposition 40. Let F F F (17) , and suppose that its connected two-point function is, off the diagonal, a locally integrable function S 2 ( x , y ) S_2(x,y) S 2 ( x , y ) S 2 ( x , y ) ≥ c ∣ x − y ∣ − 2 Δ ( log 1 ∣ x − y ∣ ) − k S_2(x,y)\geq c|x-y|^{-2\Delta}\left(\log\frac{1}{|x-y|}\right)^{-k} S 2 ( x , y ) ≥ c ∣ x − y ∣ − 2Δ ( log ∣ x − y ∣ 1 ) − k 0 < ∣ x − y ∣ ≤ r 0 0<|x-y|\leq r_0 0 < ∣ x − y ∣ ≤ r 0 c > 0 c>0 c > 0 k ≥ 0 k\geq0 k ≥ 0 2 Δ > d 2\Delta>d 2Δ > d F F F
Proof. Applying (16)`` to χ \chi χ − χ -\chi − χ cosh x ≥ 1 + x 2 2 \cosh x\geq1+\frac{x^2}{2} cosh x ≥ 1 + 2 x 2 E F ( χ ) 2 ≤ 2 ( e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) − 1 ) \mathbb{E}F(\chi)^2\leq2(e^{C(\|\chi\|_1+\|\chi\|_2^2)}-1) E F ( χ ) 2 ≤ 2 ( e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) − 1 ) φ , ψ ∈ C c ∞ ( R d ) \varphi,\psi\in C_c^\infty(\mathbb{R}^d) φ , ψ ∈ C c ∞ ( R d ) 1 2 R \frac{1}{2}R 2 1 R δ > 0 \delta>0 δ > 0 φ ε ( x ) = ε − d / 2 φ ( x / ε ) \varphi_\varepsilon(x)=\varepsilon^{-d/2}\varphi(x/\varepsilon) φ ε ( x ) = ε − d /2 φ ( x / ε ) ψ ε \psi_\varepsilon ψ ε ∥ φ ε ∥ 2 = ∥ φ ∥ 2 \|\varphi_\varepsilon\|_2=\|\varphi\|_2 ∥ φ ε ∥ 2 = ∥ φ ∥ 2 ∥ φ ε ∥ 1 = ε d / 2 ∥ φ ∥ 1 ≤ ∥ φ ∥ 1 \|\varphi_\varepsilon\|_1=\varepsilon^{d/2}\|\varphi\|_1\leq\|\varphi\|_1 ∥ φ ε ∥ 1 = ε d /2 ∥ φ ∥ 1 ≤ ∥ φ ∥ 1 ε ≤ 1 \varepsilon\leq1 ε ≤ 1 F ( φ ε ) F(\varphi_\varepsilon) F ( φ ε ) F ( ψ ε ) F(\psi_\varepsilon) F ( ψ ε ) ε \varepsilon ε φ ε \varphi_\varepsilon φ ε ψ ε \psi_\varepsilon ψ ε u , v u,v u , v ε δ ≤ ∣ u − v ∣ ≤ ε R \varepsilon\delta\leq|u-v|\leq\varepsilon R ε δ ≤ ∣ u − v ∣ ≤ εR ε R ≤ r 0 \varepsilon R\leq r_0 εR ≤ r 0 ε R < 1 \varepsilon R<1 εR < 1 S 2 S_2 S 2
Cov ( F ( φ ε ) , F ( ψ ε ) ) = ε d ∬ φ ( u ) ψ ( v ) S 2 ( ε u , ε v ) d u d v ≥ c R − 2 Δ ε d − 2 Δ ( log 1 ε δ ) − k ∥ φ ∥ 1 ∥ ψ ∥ 1 , \begin{aligned}
\operatorname{Cov}(F(\varphi_\varepsilon),F(\psi_\varepsilon))
=\varepsilon^d\iint\varphi(u)\psi(v)S_2(\varepsilon u,\varepsilon v)\,du\,dv \\
&\geq cR^{-2\Delta}\varepsilon^{d-2\Delta}\left(\log\frac{1}{\varepsilon\delta}\right)^{-k}\|\varphi\|_1\|\psi\|_1,
\end{aligned} Cov ( F ( φ ε ) , F ( ψ ε )) = ε d ∬ φ ( u ) ψ ( v ) S 2 ( ε u , ε v ) d u d v ≥ c R − 2Δ ε d − 2Δ ( log ε δ 1 ) − k ∥ φ ∥ 1 ∥ ψ ∥ 1 , which tends to infinity as ε → 0 \varepsilon\to0 ε → 0
The action density Tr F μ ν F μ ν \operatorname{Tr}F_{\mu\nu}F_{\mu\nu} Tr F μν F μν Δ = 4 \Delta=4 Δ = 4 ∣ x − y ∣ − 8 |x-y|^{-8} ∣ x − y ∣ − 8 14 [74 ] states for the smeared action density, has 2 Δ = 8 > 4 2\Delta=8>4 2Δ = 8 > 4 k = 2 k=2 k = 2 (17) , and rotation invariance for it has to come from another source, as
g ( a ) − 2 = 2 b 0 log 1 a Λ + O ( log log 1 a Λ ) , g(a)^{-2}=2b_0\log\frac{1}{a\Lambda}+O\left(\log\log\frac{1}{a\Lambda}\right), g ( a ) − 2 = 2 b 0 log a Λ 1 + O ( log log a Λ 1 ) , so the inverse coupling β W = 2 N / g ( a ) 2 \beta_{\mathrm W}=2N/g(a)^2 β W = 2 N / g ( a ) 2 11 N 2 12 π 2 log 1 a Λ \frac{11N^2}{12\pi^2}\log\frac{1}{a\Lambda} 12 π 2 11 N 2 log a Λ 1 [74 ] , where it is reduced to a separate infrared hypothesis. In three dimensions the free part of the action density already has 2 Δ = 6 > 3 2\Delta=6>3 2Δ = 6 > 3 (17) suits fields of dimension at most d / 2 d/2 d /2
Positivity, symmetry and regularity do not produce a gap by themselves.
Proposition 41. On R 4 \mathbb{R}^{4} R 4 F F F
C ^ ( p ) = ∫ 0 1 m 2 ∣ p ∣ 2 + m 2 d m . \widehat{C}(p)=\int_{0}^{1}\frac{m^{2}}{|\mathbf{p}|^{2}+m^{2}}\,dm. C ( p ) = ∫ 0 1 ∣ p ∣ 2 + m 2 m 2 d m . Then F F F (17) with C = 1 2 C=\frac{1}{2} C = 2 1
Proof. Since 0 ≤ C ^ ≤ 1 0\leq\widehat{C}\leq1 0 ≤ C ≤ 1 E e F ( χ ) = e 1 2 ⟨ χ , C χ ⟩ ≤ e 1 2 ∥ χ ∥ 2 \mathbb{E}e^{F(\chi)}=e^{\frac{1}{2}\langle\chi,C\chi\rangle}\leq e^{\frac{1}{2}\|\chi\|^{2}} E e F ( χ ) = e 2 1 ⟨ χ , C χ ⟩ ≤ e 2 1 ∥ χ ∥ 2 m ∈ ( 0 , 1 ) m\in(0,1) m ∈ ( 0 , 1 ) m 2 d m m^{2}dm m 2 d m m m m [39 Sect. 6.2] ; a positive combination of reflection positive covariances is reflection positive, and so is the Gaussian measure with such a covariance [39 Sect. 6.2] . For χ ( t , x ) = h ( t ) g ( x ) \chi(t,x)=h(t)g(x) χ ( t , x ) = h ( t ) g ( x ) h ≥ 0 h\geq0 h ≥ 0 t > 0 t>0 t > 0 g ^ ( 0 ) ≠ 0 \widehat{g}(0)\ne0 g ( 0 ) = 0 v = F ( χ ) v=F(\chi) v = F ( χ ) x x x
⟨ v , e − s H v ⟩ = ∫ 0 1 m 2 d m ∫ ∣ g ^ ( k ) ∣ 2 2 ω e − ω s ( ∫ 0 ∞ h ( t ) e − ω t d t ) 2 d k , \langle v,e^{-sH}v\rangle=\int_{0}^{1}m^{2}\,dm\int\frac{|\widehat{g}(k)|^{2}}{2\omega}e^{-\omega s}\left(\int_{0}^{\infty}h(t)e^{-\omega t}\,dt\right)^{2}dk, ⟨ v , e − sH v ⟩ = ∫ 0 1 m 2 d m ∫ 2 ω ∣ g ( k ) ∣ 2 e − ω s ( ∫ 0 ∞ h ( t ) e − ω t d t ) 2 d k , with ω = ∣ k ∣ 2 + m 2 \omega=\sqrt{|\mathbf{k}|^{2}+m^{2}} ω = ∣ k ∣ 2 + m 2 g ^ \widehat{g} g H H H m 0 > 0 m_{0}>0 m 0 > 0 e − m 0 s ∥ v ∥ 2 e^{-m_{0}s}\|v\|^{2} e − m 0 s ∥ v ∥ 2 0 < η < m 0 0<\eta<m_{0} 0 < η < m 0 m , ∣ k ∣ ≤ η / 2 m,|\mathbf{k}|\leq\eta/2 m , ∣ k ∣ ≤ η /2 c η e − η s c_{\eta}e^{-\eta s} c η e − η s c η > 0 c_{\eta}>0 c η > 0 s → ∞ s\to\infty s → ∞
The Gaussian field of Proposition 41 is not a gauge theory; it shows only that an infrared estimate, such as a bound ⟨ v a , T a ⌊ t / a ⌋ v a ⟩ ≤ C v e − m ∗ t \langle v_a,T_a^{\lfloor t/a\rfloor}v_a\rangle\leq C_v e^{-m_*t} ⟨ v a , T a ⌊ t / a ⌋ v a ⟩ ≤ C v e − m ∗ t a a a m ∗ m_* m ∗ 1 / a 1/a 1/ a [15 ] such bounds give trivial limits. The routes we know to a continuum gap therefore all stop at the same two estimates: the construction of the infinite-volume continuum measure past Bałaban’s small-field analysis [5 , 7 ] , which the conditional construction of [74 ] assumes clause by clause and which yields there a non-negative Hamiltonian without any statement on its spectrum, and a decay estimate in physical units at large scales. The finite-size criterion described in [74 Sect. 9.4] reduces a gap to a threshold condition on one box, uniformly in the cutoff; as stated there, the condition fails in every small box and would have to be checked at large scales, which the construction does not reach.
X The continuum step at one physical scale
The bounds of the preceding sections hold at fixed coupling and give a gap that is at least a fixed number in lattice units, and [15 ] shows that no such bound can produce a continuum gap. This section replaces the single-link condition of Proposition 11 by a condition on one cube of n 4 n^{4} n 4 [31 ] , and proves that the condition gives a gap of order 1 / n 1/n 1/ n n n n 1 / a 1/a 1/ a Theorem 45 ). The two statements that remain unproved, the cube estimate and the existence of the limit, are stated in Sec. XI.
Averaged cubes
For a finite set V ⊂ A V \subset A V ⊂ A γ V ( ⋅ ∣ Q ) \gamma_V(\cdot\mid Q) γ V ( ⋅ ∣ Q ) G V G^V G V exp ( N β ∑ p ∩ V ≠ ∅ Re Tr Q p F ) \exp\left(N\beta\sum_{p\cap V\ne\varnothing}\operatorname{Re}\operatorname{Tr}Q_p^F\right) exp ( N β ∑ p ∩ V = ∅ Re Tr Q p F ) V V V Q Q Q Q e Q_e Q e e ∈ V e\in V e ∈ V V V V γ e F \gamma_e^F γ e F d V ( Q V , Q V ′ ) = ∑ i ∈ V δ ( Q i , Q i ′ ) d_V(Q_V,Q'_V)=\sum_{i\in V}\delta(Q_i,Q'_i) d V ( Q V , Q V ′ ) = ∑ i ∈ V δ ( Q i , Q i ′ ) j ∈ A ∖ V j\in A\setminus V j ∈ A ∖ V
C V , j = sup W 1 d V ( γ V ( ⋅ ∣ Q ) , γ V ( ⋅ ∣ Q ′ ) ) δ ( Q j , Q j ′ ) . C_{V,j}=\sup\frac{W_1^{d_V}\left(\gamma_V(\cdot\mid Q),\gamma_V(\cdot\mid Q')\right)}{\delta(Q_j,Q'_j)}. C V , j = sup δ ( Q j , Q j ′ ) W 1 d V ( γ V ( ⋅ ∣ Q ) , γ V ( ⋅ ∣ Q ′ ) ) . the supremum over pairs that differ only at j j j W 1 d V W_1^{d_V} W 1 d V d V d_V d V j j j V V V
Let n ≥ 2 n\ge2 n ≥ 2 V V V ( x , μ ) (x,\mu) ( x , μ ) x ∈ [ 0 , n ) 4 x\in[0,n)^4 x ∈ [ 0 , n ) 4 2 Z 4 2\mathbb{Z}^4 2 Z 4 F F F A A A γ F \gamma^F γ F V x = V + x V_x=V+x V x = V + x x ∈ 2 Z 4 x\in2\mathbb{Z}^4 x ∈ 2 Z 4 C V x , j + x = C V , j C_{V_x,j+x}=C_{V,j} C V x , j + x = C V , j N n = ( n / 2 ) 4 N_n=(n/2)^4 N n = ( n /2 ) 4 2 Z 4 2\mathbb{Z}^4 2 Z 4 A A A c c c
K c = ∑ j ∈ c ∖ V C V , j . K_c=\sum_{j\in c\setminus V}C_{V,j}. K c = j ∈ c ∖ V ∑ C V , j . Definition 42. For 0 < α < 1 0<\alpha<1 0 < α < 1 D ( n , α ) D(n,\alpha) D ( n , α ) β \beta β K c ≤ α N n K_c\le\alpha N_n K c ≤ α N n c c c
Since ∑ x ∈ 2 Z 4 , j ∉ V x C V x , j = ∑ x C V , j − x [ j − x ∉ V ] = K c ( j ) \sum_{x\in2\mathbb{Z}^4,\ j\notin V_x}C_{V_x,j}=\sum_xC_{V,j-x}[j-x\notin V]=K_{c(j)} ∑ x ∈ 2 Z 4 , j ∈ / V x C V x , j = ∑ x C V , j − x [ j − x ∈ / V ] = K c ( j ) c ( j ) c(j) c ( j ) j j j α \alpha α Figure 8 shows the cubes.
Figure 8. (a) The cube V V V n = 4 n=4 n = 4 2 Z 4 2\mathbb{Z}^{4} 2 Z 4 j j j V V V K c K_{c} K c D ( n , α ) D(n,\alpha) D ( n , α ) N n = ( n / 2 ) 4 N_{n}=(n/2)^{4} N n = ( n /2 ) 4 j j j Theorem 45 : its physical side ℓ = n a \ell=na ℓ = na n n n log ( 1 / α ) / ( 2 n ) \log(1/\alpha)/(2n) log ( 1/ α ) / ( 2 n ) a / ℓ a/\ell a / ℓ
For n = 2 n=2 n = 2 V V V V V V ρ n = 2 n \rho_n=2n ρ n = 2 n c n c_n c n ρ n \rho_n ρ n
Theorem 43. Let β ≥ 0 \beta\ge0 β ≥ 0 D ( n , α ) D(n,\alpha) D ( n , α ) λ < log ( 1 / α ) / ρ n \lambda<\log(1/\alpha)/\rho_n λ < log ( 1/ α ) / ρ n ω \omega ω log ( 1 / α ) / ( 2 n ) \log(1/\alpha)/(2n) log ( 1/ α ) / ( 2 n )
Proof. The proof of Proposition 11 uses the coefficients C e e ′ C_{ee'} C e e ′ D ( n , α ) D(n,\alpha) D ( n , α ) λ > 0 \lambda>0 λ > 0 α e λ ρ n < 1 \alpha e^{\lambda\rho_n}<1 α e λ ρ n < 1 θ D / ( 1 − θ ) \theta D/(1-\theta) θ D / ( 1 − θ ) D ( 1 + c n e λ ρ n ) / ( 1 − θ ) D(1+c_ne^{\lambda\rho_n})/(1-\theta) D ( 1 + c n e λ ρ n ) / ( 1 − θ ) θ = α e λ ρ n \theta=\alpha e^{\lambda\rho_n} θ = α e λ ρ n γ V x \gamma_{V_x} γ V x V x V_x V x γ F \gamma^F γ F V x V_x V x P ( Λ ) P(\Lambda) P ( Λ ) λ < log ( 1 / α ) / ( 2 n ) \lambda<\log(1/\alpha)/(2n) λ < log ( 1/ α ) / ( 2 n )
The continuum limit
Let a k = 2 − k a 0 a_k=2^{-k}a_0 a k = 2 − k a 0 β k ≥ 0 \beta_k\ge0 β k ≥ 0 β k → ∞ \beta_k\to\infty β k → ∞ ω k \omega_k ω k β k \beta_k β k T k \mathsf{T}_k T k H k \mathcal{H}_k H k Ω k \Omega_k Ω k § 2.4.6 ). Let T \mathcal{T} T R 4 \mathbb{R}^4 R 4 { t > 0 } \{t>0\} { t > 0 } k k k f ↦ O k ( f ) f\mapsto O_k(f) f ↦ O k ( f ) T \mathcal{T} T O k ( f ) = Z k ∑ x a k 4 f ( a k x ) Φ ( x ) O_k(f)=Z_k\sum_xa_k^4f(a_kx)\Phi(x) O k ( f ) = Z k ∑ x a k 4 f ( a k x ) Φ ( x ) Φ \Phi Φ Z k Z_k Z k Φ ( x ) \Phi(x) Φ ( x ) Φ \Phi Φ x x x ϑ \vartheta ϑ τ t \tau_t τ t t t t
Definition 44. A continuum limit of ( ω k , O k ) (\omega_k,O_k) ( ω k , O k ) H \mathcal{H} H Ω \Omega Ω H ≥ 0 H\ge0 H ≥ 0 H Ω = 0 H\Omega=0 H Ω = 0 f ↦ ψ ( f ) f\mapsto\psi(f) f ↦ ψ ( f ) T \mathcal{T} T H \mathcal{H} H f , g ∈ T f,g\in\mathcal{T} f , g ∈ T τ ≥ 0 \tau\ge0 τ ≥ 0
⟨ ψ ( f ) , e − τ H ψ ( g ) ⟩ = lim k → ∞ ω k ( ϑ ( O k ( f ) ) τ τ / a k O k ( g ) ) , \langle\psi(f),e^{-\tau H}\psi(g)\rangle
=\lim_{k\to\infty}\omega_k\left(\vartheta(O_k(f))\,\tau_{\tau/a_k}O_k(g)\right), ⟨ ψ ( f ) , e − τ H ψ ( g )⟩ = k → ∞ lim ω k ( ϑ ( O k ( f )) τ τ / a k O k ( g ) ) , ⟨ Ω , ψ ( f ) ⟩ = lim k → ∞ ω k ( O k ( f ) ) . \langle\Omega,\psi(f)\rangle=\lim_{k\to\infty}\omega_k(O_k(f)). ⟨ Ω , ψ ( f )⟩ = k → ∞ lim ω k ( O k ( f )) . the limits taken over the k k k τ / a k \tau/a_k τ / a k H ≠ C Ω \mathcal{H}\ne\mathbb{C}\Omega H = C Ω
The Osterwalder–Schrader reconstruction [62 , 63 ] produces such a limit from convergent renormalised Schwinger functions that satisfy the axioms; the definition records only what the next theorem uses.
Theorem 45. Suppose that there are α < 1 \alpha<1 α < 1 ℓ > 0 \ell>0 ℓ > 0 n k n_k n k n k a k ≤ ℓ n_ka_k\le\ell n k a k ≤ ℓ D ( n k , α ) D(n_k,\alpha) D ( n k , α ) β k \beta_k β k k k k ( ω k , O k ) (\omega_k,O_k) ( ω k , O k )
σ ( H ) ⊂ { 0 } ∪ [ m ∗ , ∞ ) , m ∗ = log ( 1 / α ) 2 ℓ , \sigma(H)\subset\{0\}\cup[m_*,\infty),\qquad m_*=\frac{\log(1/\alpha)}{2\ell}, σ ( H ) ⊂ { 0 } ∪ [ m ∗ , ∞ ) , m ∗ = 2 ℓ log ( 1/ α ) , and 0 0 0 σ ( H ) ∩ [ m ∗ , ∞ ) ≠ ∅ \sigma(H)\cap[m_*,\infty)\ne\varnothing σ ( H ) ∩ [ m ∗ , ∞ ) = ∅ H H H m ∗ m_* m ∗
Proof. By Theorem 43 the state ω k \omega_k ω k T k \mathsf{T}_k T k log ( 1 / α ) / ( 2 n k ) \log(1/\alpha)/(2n_k) log ( 1/ α ) / ( 2 n k ) Ω k \Omega_k Ω k f ∈ T f\in\mathcal{T} f ∈ T O = O k ( f ) O=O_k(f) O = O k ( f ) φ k = Ψ O − ω k ( O ) Ω k \varphi_k=\Psi_O-\omega_k(O)\Omega_k φ k = Ψ O − ω k ( O ) Ω k § 2.4.6 , ⟨ Ψ F , τ t Ψ H ⟩ = ω k ( ϑ ( F ‾ ) τ t H ) \langle\Psi_F,\tau_t\Psi_H\rangle=\omega_k(\vartheta(\overline{F})\tau_tH) ⟨ Ψ F , τ t Ψ H ⟩ = ω k ( ϑ ( F ) τ t H ) ⟨ Ω k , Ψ O ⟩ = ω k ( O ) \langle\Omega_k,\Psi_O\rangle=\omega_k(O) ⟨ Ω k , Ψ O ⟩ = ω k ( O ) T k Ω k = Ω k \mathsf{T}_k\Omega_k=\Omega_k T k Ω k = Ω k t ≥ 0 t\ge0 t ≥ 0
⟨ φ k , T k t φ k ⟩ = ω k ( ϑ ( O ‾ ) τ t O ) − ∣ ω k ( O ) ∣ 2 ≤ e − t log ( 1 / α ) / ( 2 n k ) ∥ φ k ∥ 2 . \langle\varphi_k,\mathsf{T}_k^t\varphi_k\rangle
=\omega_k\left(\vartheta(\overline{O})\tau_tO\right)-|\omega_k(O)|^2
\le e^{-t\log(1/\alpha)/(2n_k)}\|\varphi_k\|^2. ⟨ φ k , T k t φ k ⟩ = ω k ( ϑ ( O ) τ t O ) − ∣ ω k ( O ) ∣ 2 ≤ e − t l o g ( 1/ α ) / ( 2 n k ) ∥ φ k ∥ 2 . Take τ ≥ 0 \tau\ge0 τ ≥ 0 k k k t = τ / a k t=\tau/a_k t = τ / a k t / ( 2 n k ) = τ / ( 2 n k a k ) ≥ τ / ( 2 ℓ ) t/(2n_k)=\tau/(2n_ka_k)\ge\tau/(2\ell) t / ( 2 n k ) = τ / ( 2 n k a k ) ≥ τ / ( 2 ℓ ) e − m ∗ τ ∥ φ k ∥ 2 e^{-m_*\tau}\|\varphi_k\|^2 e − m ∗ τ ∥ φ k ∥ 2 Definition 44 . With φ = ψ ( f ) − ⟨ Ω , ψ ( f ) ⟩ Ω \varphi=\psi(f)-\langle\Omega,\psi(f)\rangle\Omega φ = ψ ( f ) − ⟨ Ω , ψ ( f )⟩ Ω e − τ H Ω = Ω e^{-\tau H}\Omega=\Omega e − τ H Ω = Ω
⟨ φ , e − τ H φ ⟩ ≤ e − m ∗ τ ∥ φ ∥ 2 . \langle\varphi,e^{-\tau H}\varphi\rangle\le e^{-m_*\tau}\|\varphi\|^2. ⟨ φ , e − τ H φ ⟩ ≤ e − m ∗ τ ∥ φ ∥ 2 . The maps f ↦ O k ( f ) f\mapsto O_k(f) f ↦ O k ( f ) f ↦ ψ ( f ) f\mapsto\psi(f) f ↦ ψ ( f ) φ \varphi φ Ω ⊥ \Omega^\perp Ω ⊥ ψ \psi ψ Ω ⊥ \Omega^\perp Ω ⊥ τ \tau τ τ ≥ 0 \tau\ge0 τ ≥ 0 e − τ H e^{-\tau H} e − τ H φ ∈ Ω ⊥ \varphi\in\Omega^\perp φ ∈ Ω ⊥ μ φ \mu_\varphi μ φ H H H ∫ e − τ E d μ φ ( E ) ≤ e − m ∗ τ μ φ ( [ 0 , ∞ ) ) \int e^{-\tau E}\,\mathrm{d}\mu_\varphi(E) \le e^{-m_\ast\tau}\mu_\varphi([0,\infty)) ∫ e − τ E d μ φ ( E ) ≤ e − m ∗ τ μ φ ([ 0 , ∞ )) τ ≥ 0 \tau\ge0 τ ≥ 0 μ φ ( [ 0 , m ∗ − η ] ) > 0 \mu_\varphi([0,m_\ast-\eta])>0 μ φ ([ 0 , m ∗ − η ]) > 0 η > 0 \eta>0 η > 0 μ φ ( [ 0 , m ∗ − η ] ) e − ( m ∗ − η ) τ \mu_\varphi([0,m_\ast-\eta])e^{-(m_\ast-\eta)\tau} μ φ ([ 0 , m ∗ − η ]) e − ( m ∗ − η ) τ τ \tau τ μ φ \mu_\varphi μ φ [ m ∗ , ∞ ) [m_\ast,\infty) [ m ∗ , ∞ ) φ ∈ Ω ⊥ \varphi\in\Omega^\perp φ ∈ Ω ⊥ 0 0 0 Ω ⊥ ≠ { 0 } \Omega^\perp\ne\{0\} Ω ⊥ = { 0 } [ m ∗ , ∞ ) [m_\ast,\infty) [ m ∗ , ∞ )
The proof uses D ( n k , α ) D(n_k,\alpha) D ( n k , α )
Theorem 46. Let m 0 > 0 m_0>0 m 0 > 0 T 0 \mathcal{T}_0 T 0 T \mathcal{T} T f ∈ T 0 f \in\mathcal{T}_0 f ∈ T 0 C f < ∞ C_f<\infty C f < ∞ τ f ≥ 0 \tau_f\ge0 τ f ≥ 0 k f k_f k f
C k f ( τ ) = ω k ( ϑ ( O k ( f ) ) τ / a k O k ( f ) ) − ∣ ω k ( O k ( f ) ) ∣ 2 ≤ C f e − m 0 τ C_k^f(\tau)=\omega_k\left(\vartheta(O_k(f))_{\tau/a_k}O_k(f)\right)-\left|\omega_k(O_k(f))\right|^2
\le C_f e^{-m_0\tau} C k f ( τ ) = ω k ( ϑ ( O k ( f ) ) τ / a k O k ( f ) ) − ∣ ω k ( O k ( f )) ∣ 2 ≤ C f e − m 0 τ for all k ≥ k f k\ge k_f k ≥ k f τ ≥ τ f \tau\ge\tau_f τ ≥ τ f τ / a k ∈ Z \tau/a_k\in\mathbb{Z} τ / a k ∈ Z ( ω k , O k ) (\omega_k,O_k) ( ω k , O k ) σ ( H ) ⊂ { 0 } ∪ [ m 0 , ∞ ) \sigma(H)\subset\{0\}\cup[m_0,\infty) σ ( H ) ⊂ { 0 } ∪ [ m 0 , ∞ ) 0 0 0 H H H m 0 m_0 m 0
Proof. For f ∈ T f\in\mathcal{T} f ∈ T φ ( f ) = ψ ( f ) − ⟨ Ω , ψ ( f ) ⟩ Ω \varphi(f)=\psi(f)-\langle\Omega,\psi(f)\rangle\Omega φ ( f ) = ψ ( f ) − ⟨ Ω , ψ ( f )⟩ Ω f f f e − τ H e^{-\tau H} e − τ H Ω \Omega Ω ⟨ φ ( f ) , e − τ H φ ( f ) ⟩ = lim k C k f ( τ ) \langle\varphi(f),e^{-\tau H}\varphi(f)\rangle=\lim_k C_k^f(\tau) ⟨ φ ( f ) , e − τ H φ ( f )⟩ = lim k C k f ( τ ) f ∈ T 0 f\in\mathcal{T}_0 f ∈ T 0 μ \mu μ H H H φ ( f ) \varphi(f) φ ( f ) ∫ e − τ E d μ ( E ) ≤ C f e − m 0 τ \int e^{-\tau E}\,\mathrm{d}\mu(E)\le C_f e^{-m_0\tau} ∫ e − τ E d μ ( E ) ≤ C f e − m 0 τ τ ≥ τ f \tau\ge\tau_f τ ≥ τ f μ ( [ 0 , m 0 − η ] ) > 0 \mu([0,m_0-\eta])>0 μ ([ 0 , m 0 − η ]) > 0 η ∈ ( 0 , m 0 ) \eta\in(0,m_0) η ∈ ( 0 , m 0 ) μ ( [ 0 , m 0 − η ] ) e − ( m 0 − η ) τ \mu([0,m_0-\eta])e^{-(m_0-\eta)\tau} μ ([ 0 , m 0 − η ]) e − ( m 0 − η ) τ τ \tau τ E [ 0 , m 0 ) E_{[0,m_0)} E [ 0 , m 0 ) H H H φ ( T 0 ) \varphi(\mathcal{T}_0) φ ( T 0 ) φ ( T ) \varphi(\mathcal{T}) φ ( T ) φ ( T ) \varphi(\mathcal{T}) φ ( T ) Ω ⊥ \Omega^\perp Ω ⊥ ψ ( T ) \psi(\mathcal{T}) ψ ( T ) E [ 0 , m 0 ) = ⟨ Ω , ⋅ ⟩ Ω E_{[0,m_0)}=\langle\Omega,\cdot\rangle\Omega E [ 0 , m 0 ) = ⟨ Ω , ⋅ ⟩ Ω
Theorem 45 is the case m 0 = m ∗ m_0=m_\ast m 0 = m ∗ τ f = 0 \tau_f=0 τ f = 0 C f = sup k ≥ k f ∥ φ k ∥ 2 C_f=\sup_{k\ge k_f}\|\varphi_k\|^2 C f = sup k ≥ k f ∥ φ k ∥ 2 ∥ φ k ∥ 2 = C k f ( 0 ) \|\varphi_k\|^2=C_k^f(0) ∥ φ k ∥ 2 = C k f ( 0 ) Theorem 46 needs neither a lattice gap nor the gauge of this paper. It applies to the output of a cluster expansion of any coarse effective theory, provided the expansion bounds the correlations of a spanning set of renormalised observables in physical units.
Theorem 45 bypasses the obstruction of [15 ] : under the hypothesis the lattice gap is at least log ( 1 / α ) / ( 2 n k ) \log(1/\alpha)/(2n_k) log ( 1/ α ) / ( 2 n k ) a k / ℓ a_k/\ell a k / ℓ [15 ] the hypothesis implies the gap part (R2) of Hypothesis R, with m = m ∗ m=m_\ast m = m ∗
Smoothed observables
The observables O k ( f ) O_k(f) O k ( f ) [55 ] 15 [56 ] . Flow in all four directions mixes the two sides of the reflection plane, but the spatial flow of [67, Sec. II.B} does not: it flows the links of each time slice under the Wilson action of that slice and leaves the temporal links alone.
Fix a bi-invariant Riemannian metric on G G G U U U Z 4 \mathbb{Z}^4 Z 4 F F F 1 1 1 Φ s ( U ) \Phi_s(U) Φ s ( U ) s ∈ R s \in\mathbb{R} s ∈ R
d d s U e = − ∇ e ∑ p ∋ e , p s p a t i a l ( N − Re Tr U p ) \frac{d}{ds}U_e=-\nabla_e\sum_{\substack{p\ni e,\\p\ \mathrm{spatial}}}\left(N-\operatorname{Re}\operatorname{Tr}U_p\right) d s d U e = − ∇ e p ∋ e , p spatial ∑ ( N − Re Tr U p ) for every spatial link e e e e e e ∇ e \nabla_e ∇ e U e U_e U e
Proposition 47. (i) Φ s \Phi_s Φ s Φ s ∘ Φ s ′ = Φ s + s ′ \Phi_s\circ\Phi_{s'}=\Phi_{s+s'} Φ s ∘ Φ s ′ = Φ s + s ′ Φ s \Phi_s Φ s Φ s ( U ) e \Phi_s(U)_e Φ s ( U ) e e e e κ , c \kappa,c κ , c G G G Φ s R \Phi_s^R Φ s R R R R e e e
δ ( Φ s ( U ) e , Φ s R ( U ) e ) ≤ D e κ ∣ s ∣ ( c ∣ s ∣ ) R + 1 ( R + 1 ) ! \delta\left(\Phi_s(U)_e,\Phi_s^R(U)_e\right)\leq De^{\kappa|s|}\frac{(c|s|)^{R+1}}{(R+1)!} δ ( Φ s ( U ) e , Φ s R ( U ) e ) ≤ D e κ ∣ s ∣ ( R + 1 )! ( c ∣ s ∣ ) R + 1 for every configuration U U U
(ii) Let ω \omega ω § 2.4.6 and O O O [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ] O ∘ Φ s O\circ\Phi_s O ∘ Φ s [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ]
(iii) Theorem 45 holds with O k ( f ) O_k(f) O k ( f ) O k ( f ) ∘ Φ s k O_k(f)\circ\Phi_{s_k} O k ( f ) ∘ Φ s k s k ∈ R s_k\in\mathbb{R} s k ∈ R s k = τ / a k 2 s_k=\tau/a_k^2 s k = τ / a k 2 τ \tau τ
(iv) The image measure ( Φ s ) ∗ ω (\Phi_s)_*\omega ( Φ s ) ∗ ω ω \omega ω
Proof. (i) Embed G G G e e e G G G U e U_e U e G G G G G G G G G s s s d e ′ ( s ) d_{e'}(s) d e ′ ( s ) e ′ e' e ′ κ 0 , c 1 \kappa_0,c_1 κ 0 , c 1 d e ′ ( s ) ≤ ∫ 0 ∣ s ∣ ( κ 0 d e ′ + c 1 ∑ e ′ ′ ∼ e ′ d e ′ ′ ) d_{e'}(s)\leq\int_0^{|s|}(\kappa_0d_{e'}+c_1\sum_{e''\sim e'}d_{e''}) d e ′ ( s ) ≤ ∫ 0 ∣ s ∣ ( κ 0 d e ′ + c 1 ∑ e ′′ ∼ e ′ d e ′′ ) e ′ e' e ′ R R R e e e e ′ ′ ∼ e ′ e''\sim e' e ′′ ∼ e ′ d e ′ ≤ D d_{e'}\leq D d e ′ ≤ D R R R R + 1 R+1 R + 1 ∼ \sim ∼ ∑ k > R x k / k ! ≤ e x x R + 1 / ( R + 1 ) ! \sum_{k>R}x^k/k!\leq e^x x^{R+1}/(R+1)! ∑ k > R x k / k ! ≤ e x x R + 1 / ( R + 1 )! κ = 12 c 1 \kappa=12c_1 κ = 12 c 1 κ = κ 0 + c \kappa=\kappa_0+c κ = κ 0 + c
(ii) The truncated flow Φ s R \Phi_s^R Φ s R Φ s \Phi_s Φ s O ∘ Φ s R O\circ\Phi_s^R O ∘ Φ s R [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ] O O O O ∘ Φ s O\circ\Phi_s O ∘ Φ s R → ∞ R\to\infty R → ∞
(iii) The proof of Theorem 45 uses O k ( f ) O_k(f) O k ( f ) Ψ O k ( f ) \Psi_{O_k(f)} Ψ O k ( f ) ⟨ Ψ F , T H Ψ H ⟩ = ω ( ϑ ( F ‾ ) τ t ( H ) ) \langle\Psi_F,\mathsf{T}_H\Psi_H\rangle=\omega(\vartheta(\overline{F})\tau_t(H)) ⟨ Ψ F , T H Ψ H ⟩ = ω ( ϑ ( F ) τ t ( H )) § 2.4.6 , both extend to uniform limits of local functions at positive times, and f ↦ O k ( f ) ∘ Φ s k f\mapsto O_k(f)\circ\Phi_{s_k} f ↦ O k ( f ) ∘ Φ s k
(iv) By (i), Φ s \Phi_s Φ s Φ − s \Phi_{-s} Φ − s ϑ \vartheta ϑ τ t \tau_t τ t ( Φ s ) ∗ ω ( ϑ ( F ‾ ) τ t H ) = ω ( ϑ ( F ∘ Φ s ‾ ) τ t ( H ∘ Φ s ) ) (\Phi_s)_*\omega(\vartheta(\overline{F})\tau_tH)=\omega(\vartheta(\overline{F\circ\Phi_s})\tau_t(H\circ\Phi_s)) ( Φ s ) ∗ ω ( ϑ ( F ) τ t H ) = ω ( ϑ ( F ∘ Φ s ) τ t ( H ∘ Φ s )) t = 0 t=0 t = 0 H = F H=F H = F F ↦ F ∘ Φ s F\mapsto F\circ\Phi_s F ↦ F ∘ Φ s Φ − s \Phi_{-s} Φ − s
Part (iv) says that smoothing changes the variables and not the theory: it neither creates nor removes a gap. Its use is in the choice of the observables. With O k ( f ) O_k(f) O k ( f ) ∑ x ∈ a k Z 4 a k 4 f ( a k x ) ( E τ ( x ) − ω k ( E τ ( x ) ) ) \sum_{x\in a_k\mathbb{Z}^4}a_k^4f(a_kx)(\mathcal{E}_\tau(x)-\omega_k(\mathcal{E}_\tau(x))) ∑ x ∈ a k Z 4 a k 4 f ( a k x ) ( E τ ( x ) − ω k ( E τ ( x ))) E τ ( x ) \mathcal{E}_\tau(x) E τ ( x ) [56 ] , and Theorem 45 applies to them. Theorem 48 below constructs their continuum limits along subsequences; their nontriviality is not proved here.
Renormalised flowed observables
The spatially flowed fields can be renormalised by one condition at a single physical separation, and their correlation functions then have continuum limits along subsequences without further assumptions. What remains open is whether such a limit is trivial, and that question reduces to one ratio of lattice two-point functions.
Fix a bounded gauge-invariant function Φ \Phi Φ { x 0 = 0 } \{x_0=0\} { x 0 = 0 } τ > 0 \tau>0 τ > 0 f ∈ C c ( R 4 ) f\in C_c(\mathbb{R}^4) f ∈ C c ( R 4 ) { t > 0 } \{t>0\} { t > 0 } s k = τ / a k 2 s_k=\tau/a_k^2 s k = τ / a k 2 Φ x \Phi_x Φ x Φ \Phi Φ x ∈ Z 4 x\in\mathbb{Z}^4 x ∈ Z 4
A k = ∑ x ∈ Z 4 a k 4 f ( a k x ) ( Φ x ∘ Φ s k − ω k ( Φ x ∘ Φ s k ) ) , A_k=\sum_{x\in\mathbb{Z}^4}a_k^4f(a_kx)\left(\Phi_x\circ\Phi_{s_k}-\omega_k(\Phi_x\circ\Phi_{s_k})\right), A k = x ∈ Z 4 ∑ a k 4 f ( a k x ) ( Φ x ∘ Φ s k − ω k ( Φ x ∘ Φ s k ) ) , C k ( σ ) = ω k ( ϑ ( A k ‾ ) τ σ / a k A k ) C_k(\sigma)=\omega_k\left(\vartheta(\overline{A_k})\tau_{\sigma/a_k}A_k\right) C k ( σ ) = ω k ( ϑ ( A k ) τ σ / a k A k ) for dyadic σ ≥ 0 \sigma\geq0 σ ≥ 0 σ / a k \sigma/a_k σ / a k k k k C k ( 0 ) ≥ 0 C_k(0)\geq0 C k ( 0 ) ≥ 0 C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0 Z k = C k ( 0 ) − 1 / 2 Z_k=C_k(0)^{-1/2} Z k = C k ( 0 ) − 1/2 D \mathcal{D} D y = ( t , y ′ ) ∈ R 4 y=(t,y')\in\mathbb{R}^4 y = ( t , y ′ ) ∈ R 4 a 0 a_0 a 0 t > 0 t>0 t > 0 y ∈ D y \in\mathcal{D} y ∈ D k k k y / a k ∈ Z 4 y/a_k \in\mathbb{Z}^4 y / a k ∈ Z 4 A k y A_k^y A k y A k A_k A k y / a k y/a_k y / a k
Theorem 48. (i) For y , y ′ ∈ D y,y' \in\mathcal{D} y , y ′ ∈ D k k k y / a k , y ′ / a k ∈ Z 4 y/a_k,y'/a_k \in\mathbb{Z}^4 y / a k , y ′ / a k ∈ Z 4
Z k 2 ∣ ω k ( ϑ ( A k y ‾ ) A k y ′ ) ∣ ≤ 1. Z_k^2\left|\omega_k\left(\vartheta(\overline{A_k^y})A_k^{y'}\right)\right| \le1. Z k 2 ω k ( ϑ ( A k y ) A k y ′ ) ≤ 1. (ii) Every sequence of k k k K ( y , y ′ ) = lim Z k 2 ω k ( ϑ ( A k y ‾ ) A k y ′ ) K(y,y')=\lim Z_k^2\omega_k\left(\vartheta(\overline{A_k^y})A_k^{y'}\right) K ( y , y ′ ) = lim Z k 2 ω k ( ϑ ( A k y ) A k y ′ ) exists for all y , y ′ ∈ D y,y'\in\mathcal{D} y , y ′ ∈ D and ρ ( σ ) = lim C k ( σ ) / C k ( 0 ) \rho(\sigma)=\lim C_k(\sigma)/C_k(0) ρ ( σ ) = lim C k ( σ ) / C k ( 0 ) exists for all dyadic σ > 0 \sigma>0 σ > 0 Along it there is a probability measure μ \mu μ on [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] with
ρ ( σ ) = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) ( σ > 0 ) , \rho(\sigma)=\int_{[0,\infty]}e^{-\sigma E}\,d\mu(E) \qquad(\sigma>0), ρ ( σ ) = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) ( σ > 0 ) , the integrand being 0 at E = ∞ E=\infty E = ∞
(iii) Along such a subsequence, let T \mathcal{T} T be spanned by the translates f 0 ( ⋅ − y ) f_0(\mathord\cdot-y) f 0 ( ⋅ − y ) y ∈ D y\in\mathcal{D} y ∈ D and one more function f ∗ ∈ C c ( R 4 ) f_*\in C_c(\mathbb{R}^4) f ∗ ∈ C c ( R 4 ) supported in { t > 0 } \{t>0\} { t > 0 } outside their span, and let O k O_k O k be the linear map with O k ( f 0 ( ⋅ − y ) ) = Z k A k y O_k(f_0(\mathord\cdot-y))=Z_kA_k^y O k ( f 0 ( ⋅ − y )) = Z k A k y and O k ( f ∗ ) = 1 O_k(f_*)=1 O k ( f ∗ ) = 1 Then ( ω k , O k ) (\omega_k,O_k) ( ω k , O k ) has a continuum limit in the sense of Definition 44 , on which the dyadic spatial translations act by unitary operators commuting with H H H
(iv) This limit is nontrivial if and only if μ ( [ 0 , ∞ ) ) > 0 \mu([0,\infty))>0 μ ([ 0 , ∞ )) > 0 if and only if ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 for one dyadic σ > 0 \sigma>0 σ > 0 and then ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 for all of them. Consequently a nontrivial limit exists along some subsequence if and only if lim sup k C k ( σ ) / C k ( 0 ) > 0 \limsup_{k}C_k(\sigma)/C_k(0)>0 lim sup k C k ( σ ) / C k ( 0 ) > 0 for some dyadic σ > 0 \sigma>0 σ > 0
(v) If the hypothesis of Theorem 45 holds, then μ \mu μ is carried by [ m ∗ , ∞ ] [m_*,\infty] [ m ∗ , ∞ ] and ρ ( σ ) ≤ e − m ∗ σ \rho(\sigma)\le e^{-m_*\sigma} ρ ( σ ) ≤ e − m ∗ σ
Proof. By Proposition 47 (ii) each Φ x ∘ Φ s k \Phi_x\circ\Phi_{s_k} Φ x ∘ Φ s k A k y A_k^y A k y ≥ 1 \ge1 ≥ 1 § 2.4.6 applies to them (its footnote). Write Ψ k = Z k Ψ A k \Psi_k=Z_k\Psi_{A_k} Ψ k = Z k Ψ A k ⟨ Ω k , Ψ A k ⟩ = ω k ( A k ) = 0 \langle\Omega_k,\Psi_{A_k}\rangle=\omega_k(A_k)=0 ⟨ Ω k , Ψ A k ⟩ = ω k ( A k ) = 0 ∥ Ψ k ∥ 2 = Z k 2 C k ( 0 ) = 1 \|\Psi_k\|^2=Z_k^2C_k(0)=1 ∥ Ψ k ∥ 2 = Z k 2 C k ( 0 ) = 1 Ψ k \Psi_k Ψ k Ω k \Omega_k Ω k ⟨ Ψ F , T k t Ψ H ⟩ = ω k ( ϑ ( F ‾ ) τ t H ) \langle\Psi_F,T_k^t\Psi_H\rangle=\omega_k(\vartheta(\overline{F})\tau_tH) ⟨ Ψ F , T k t Ψ H ⟩ = ω k ( ϑ ( F ) τ t H ) F F F Ψ τ t H = T k t Ψ H \Psi_{\tau_tH}=T_k^t\Psi_H Ψ τ t H = T k t Ψ H ϑ \vartheta ϑ τ t \tau_t τ t ω k \omega_k ω k U k ( x ) U_k(x) U k ( x ) T k T_k T k y = ( t , y ′ ) y=(t,y') y = ( t , y ′ )
Z k Ψ A k y = U k ( y ′ / a k ) T k t / a k Ψ k . Z_k\Psi_{A_k^y}=U_k(y'/a_k)T_k^{t/a_k}\Psi_k. Z k Ψ A k y = U k ( y ′ / a k ) T k t / a k Ψ k . (i) The left side of the inequality is ∣ ⟨ Z k Ψ A k y , Z k Ψ A k y ′ ⟩ ∣ ≤ 1 \left|\langle Z_k\Psi_{A_k^y},Z_k\Psi_{A_k^{y'}}\rangle\right|\le1 ⟨ Z k Ψ A k y , Z k Ψ A k y ′ ⟩ ≤ 1 ∥ U k ∥ = 1 \|U_k\|=1 ∥ U k ∥ = 1 0 ≤ T k ≤ 1 0\le T_k\le1 0 ≤ T k ≤ 1
(ii) The set D × D \mathcal{D}\times\mathcal{D} D × D K K K ρ ( σ ) \rho(\sigma) ρ ( σ ) C k ( σ ) / C k ( 0 ) = ⟨ Ψ k , T k σ / a k Ψ k ⟩ C_k(\sigma)/C_k(0)=\langle\Psi_k,T_k^{\sigma/a_k}\Psi_k\rangle C k ( σ ) / C k ( 0 ) = ⟨ Ψ k , T k σ / a k Ψ k ⟩ K K K y = y ′ = ( σ / 2 , 0 ) y=y'=(\sigma/2,0) y = y ′ = ( σ /2 , 0 ) ν k \nu_k ν k T k T_k T k Ψ k \Psi_k Ψ k [ 0 , 1 ] [0,1] [ 0 , 1 ] μ k \mu_k μ k λ ↦ − a k − 1 log λ ∈ [ 0 , ∞ ] \lambda\mapsto-a_k^{-1}\log\lambda\in[0,\infty] λ ↦ − a k − 1 log λ ∈ [ 0 , ∞ ] C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ k ( E ) C_k(\sigma)/C_k(0)=\int e^{-\sigma E}\,d\mu_k(E) C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ k ( E ) [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] μ k → μ \mu_k\to\mu μ k → μ E ↦ e − σ E E\mapsto e^{-\sigma E} E ↦ e − σ E [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] σ > 0 \sigma>0 σ > 0
(iii) A finite combination ∑ j c j f 0 ( ⋅ − y j ) \sum_j c_jf_0(\mathord\cdot-y_j) ∑ j c j f 0 ( ⋅ − y j ) y j y_j y j f ^ 0 ( p ) ∑ j c j e − i p ⋅ y j = 0 \widehat f_0(p)\sum_jc_je^{-ip\cdot y_j}=0 f 0 ( p ) ∑ j c j e − i p ⋅ y j = 0 f ^ 0 \widehat f_0 f 0 c j c_j c j O k O_k O k V V V v = ∑ y c y δ y v=\sum_y c_y\delta_y v = ∑ y c y δ y D \mathcal{D} D B ( v , w ) = ∑ y , y ′ c y ‾ c y ′ ′ K ( y , y ′ ) B(v,w)=\sum_{y,y'}\overline{c_y}c'_{y'}K(y,y') B ( v , w ) = ∑ y , y ′ c y c y ′ ′ K ( y , y ′ ) ⟨ v k , w k ⟩ \langle v_k,w_k\rangle ⟨ v k , w k ⟩ v k = ∑ y c y Z k Ψ A k y v_k=\sum_yc_yZ_k\Psi_{A_k^y} v k = ∑ y c y Z k Ψ A k y H 0 \mathcal{H}_0 H 0 V V V σ ≥ 0 \sigma\ge0 σ ≥ 0 x x x T ( σ ) δ y = δ y + ( σ , 0 ) T(\sigma)\delta_y=\delta_{y+(\sigma,0)} T ( σ ) δ y = δ y + ( σ , 0 ) U ( x ) δ y = δ y + ( 0 , x ) U(x)\delta_y=\delta_{y+(0,x)} U ( x ) δ y = δ y + ( 0 , x ) k k k T k σ / a k T_k^{\sigma/a_k} T k σ / a k U k ( x / a k ) U_k(x/a_k) U k ( x / a k ) B ( T ( σ ) v , w ) = B ( v , T ( σ ) w ) B(T(\sigma)v,w)=B(v,T(\sigma)w) B ( T ( σ ) v , w ) = B ( v , T ( σ ) w ) B ( T ( σ ) v , T ( σ ) w ) ≤ B ( v , w ) B(T(\sigma)v,T(\sigma)w)\le B(v,w) B ( T ( σ ) v , T ( σ ) w ) ≤ B ( v , w ) B ( v , T ( σ ) v ) ≥ 0 B(v,T(\sigma)v)\ge0 B ( v , T ( σ ) v ) ≥ 0 U ( x ) U(x) U ( x ) B B B T ( σ ) T(\sigma) T ( σ ) H 0 \mathcal{H}_0 H 0 T ( σ ) T ( σ ′ ) = T ( σ + σ ′ ) T(\sigma)T(\sigma')=T(\sigma+\sigma') T ( σ ) T ( σ ′ ) = T ( σ + σ ′ ) U ( x ) U(x) U ( x ) σ > 0 \sigma>0 σ > 0 T ( σ ) = T ( σ / 2 ) 2 T(\sigma)=T(\sigma/2)^2 T ( σ ) = T ( σ /2 ) 2 T ( σ / 2 ) T(\sigma/2) T ( σ /2 ) δ y \delta_y δ y y = ( t , y ′ ) y=(t,y') y = ( t , y ′ ) T ( t / 2 ) δ ( t / 2 , y ′ ) T(t/2)\delta_{(t/2,y')} T ( t /2 ) δ ( t /2 , y ′ ) T ( σ ) T(\sigma) T ( σ ) H 0 \mathcal{H}_0 H 0 H 0 = − log T ( 1 ) H_0=-\log T(1) H 0 = − log T ( 1 ) H 0 ≥ 0 H_0\ge0 H 0 ≥ 0 e − σ H 0 = T ( σ ) e^{-\sigma H_0}=T(\sigma) e − σ H 0 = T ( σ ) σ \sigma σ H = C Ω ⊕ H 0 \mathcal{H}=\mathbb{C}\Omega\oplus\mathcal{H}_0 H = C Ω ⊕ H 0 H = 0 ⊕ H 0 H=0\oplus H_0 H = 0 ⊕ H 0 Ψ ( f 0 ( ⋅ − y ) ) = δ y \Psi(f_0(\mathord\cdot-y))=\delta_y Ψ ( f 0 ( ⋅ − y )) = δ y Ψ ( f ∗ ) = 1 \Psi(f_*)=1 Ψ ( f ∗ ) = 1 Ψ \Psi Ψ Definition 44 hold: for two translates they are the limits defining K K K τ σ / a k A k y = A k y + ( σ , 0 ) \tau_{\sigma/a_k}A_k^y=A_k^{y+(\sigma,0)} τ σ / a k A k y = A k y + ( σ , 0 ) f ∗ f_* f ∗ ω k ( 1 ) = 1 \omega_k(1)=1 ω k ( 1 ) = 1 ω k ( ϑ ( A k y ‾ ) ) = ω k ( τ σ / a k A k y ) = 0 \omega_k(\vartheta(\overline{A_k^y}))=\omega_k(\tau_{\sigma/a_k}A_k^y)=0 ω k ( ϑ ( A k y )) = ω k ( τ σ / a k A k y ) = 0
(iv) The limit is nontrivial exactly when H 0 ≠ 0 \mathcal{H}_0\ne0 H 0 = 0 ∥ δ y ∥ 2 = K ( y , y ) = ρ ( 2 t ) \|\delta_y\|^2=K(y,y)=\rho(2t) ∥ δ y ∥ 2 = K ( y , y ) = ρ ( 2 t ) y = ( t , y ′ ) y=(t,y') y = ( t , y ′ ) ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 σ > 0 \sigma>0 σ > 0 μ ( [ 0 , ∞ ) ) > 0 \mu([0,\infty))>0 μ ([ 0 , ∞ )) > 0 ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 σ > 0 \sigma>0 σ > 0 C k ( σ ) / C k ( 0 ) C_k(\sigma)/C_k(0) C k ( σ ) / C k ( 0 )
(v) Under that hypothesis the proof of Theorem 45 gives ν k ( [ 0 , e − log ( 1 / α ) / ( 2 n k ) ] ) = 1 \nu_k([0,e^{-\log(1/\alpha)/(2n_k)}])=1 ν k ([ 0 , e − l o g ( 1/ α ) / ( 2 n k ) ]) = 1 Ψ k ⊥ Ω k \Psi_k\perp\Omega_k Ψ k ⊥ Ω k μ k \mu_k μ k [ log ( 1 / α ) / ( 2 n k a k ) , ∞ ] ⊂ [ m ∗ , ∞ ] [\log(1/\alpha)/(2n_ka_k),\infty]\subset[m_*,\infty] [ log ( 1/ α ) / ( 2 n k a k ) , ∞ ] ⊂ [ m ∗ , ∞ ] μ \mu μ □ \square □
The measure μ \mu μ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ∞ \infty ∞
lim sup k → ∞ C k ( σ ) C k ( 0 ) > 0 \limsup_{k\to\infty}\frac{C_k(\sigma)}{C_k(0)}>0 k → ∞ lim sup C k ( 0 ) C k ( σ ) > 0 for one physical separation σ \sigma σ C k ( 0 ) C_k(0) C k ( 0 ) m > 0 m>0 m > 0 e − m σ / a k e^{-m\sigma/a_k} e − mσ / a k β k → ∞ \beta_k\to\infty β k → ∞ Z k Z_k Z k C k ( 0 ) C_k(0) C k ( 0 ) f 0 f_0 f 0 Λ \Lambda Λ 0 0 0 ( 1 − ( 1 + Λ t ) e − Λ t ) / ( 4 π 2 t 2 ) (1-(1+\Lambda t)e^{-\Lambda t})/(4\pi^2t^2) ( 1 − ( 1 + Λ t ) e − Λ t ) / ( 4 π 2 t 2 ) t > 0 t>0 t > 0 Λ 2 / ( 8 π 2 ) \Lambda^2/(8\pi^2) Λ 2 / ( 8 π 2 ) 2 ( 1 − ( 1 + Λ t ) e − Λ t ) / ( Λ t ) 2 2(1-(1+\Lambda t)e^{-\Lambda t})/(\Lambda t)^2 2 ( 1 − ( 1 + Λ t ) e − Λ t ) / ( Λ t ) 2 Λ → ∞ \Lambda\to\infty Λ → ∞ f 0 f_0 f 0 Theorem 48 (iv) is about the smeared packets A k A_k A k
!The packet f 0 f_0 f 0 [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ] t = 0 t=0 t = 0 ϑ f 0 \vartheta f_0 ϑ f 0 τ σ f 0 \tau_\sigma f_0 τ σ f 0 σ \sigma σ C k ( σ ) = ⟨ Ψ A k , Π k σ / a k Ψ A k ⟩ C_k(\sigma)=\langle\Psi_{A_k},\Pi_k^{\sigma/a_k}\Psi_{A_k}\rangle C k ( σ ) = ⟨ Ψ A k , Π k σ / a k Ψ A k ⟩ ϑ f 0 \vartheta f_0 ϑ f 0 σ \sigma σ bbox=0.30,0.10,0.70,0.43
Figure 9. The packet f 0 f_0 f 0 Theorem 48 , supported in the time window [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ] t = 0 t=0 t = 0 ϑ f 0 \vartheta f_0 ϑ f 0 τ σ f 0 \tau_\sigma f_0 τ σ f 0 σ \sigma σ C k ( σ ) C_k(\sigma) C k ( σ ) C k ( 0 ) C_k(0) C k ( 0 )
When the ratio is bounded below, renormalising at the separation σ \sigma σ 0 0 0 μ k \mu_k μ k Theorem 48 .
Proposition 49. (i) Suppose C k ( σ ) ≥ c C k ( 0 ) C_k(\sigma)\ge cC_k(0) C k ( σ ) ≥ c C k ( 0 ) σ > 0 \sigma>0 σ > 0 c > 0 c>0 c > 0 k k k μ k ′ \mu'_k μ k ′ E = − a k − 1 log λ E=-a_k^{-1}\log\lambda E = − a k − 1 log λ C k ( σ ) − 1 / 2 Ψ A k ′ C_k(\sigma)^{-1/2}\Psi_{A'_k} C k ( σ ) − 1/2 Ψ A k ′ A k ′ = τ σ / 2 a k A k A'_k=\tau_{\sigma/2a_k}A_k A k ′ = τ σ /2 a k A k μ k ′ \mu'_k μ k ′ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
μ k ′ ( [ Λ , ∞ ) ) ≤ c − 1 e − σ Λ , ∫ E m d μ k ′ ≤ c − 1 ( m e σ ) m , \mu'_k([\Lambda,\infty))\le c^{-1}e^{-\sigma\Lambda},\qquad\int E^m\,d\mu'_k\le c^{-1}\left(\frac{m}{e\sigma}\right)^m, μ k ′ ([ Λ , ∞ )) ≤ c − 1 e − σ Λ , ∫ E m d μ k ′ ≤ c − 1 ( e σ m ) m , and t ↦ C k ( σ + t ) / C k ( σ ) t\mapsto C_k(\sigma+t)/C_k(\sigma) t ↦ C k ( σ + t ) / C k ( σ ) [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) ( e σ c ) − 1 (e\sigma c)^{-1} ( e σ c ) − 1 μ k ′ \mu'_k μ k ′ [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
(ii) Suppose ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ C a k \langle\Psi_k,(1-T_k)\Psi_k\rangle\le Ca_k ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ C a k C C C k k k C k ( σ ) / C k ( 0 ) ≥ e − 2 C σ C_k(\sigma)/C_k(0)\ge e^{-2C\sigma} C k ( σ ) / C k ( 0 ) ≥ e − 2 C σ C a k ≤ 1 2 Ca_k\le\frac12 C a k ≤ 2 1 σ / a k \sigma/a_k σ / a k μ k ( [ Λ , ∞ ) ) ≤ C ( a k + Λ − 1 ) \mu_k([\Lambda,\infty))\le C(a_k+\Lambda^{-1}) μ k ([ Λ , ∞ )) ≤ C ( a k + Λ − 1 )
Proof. (i) We have Ψ A k ′ = Π k σ / 2 a k Ψ A k \Psi_{A'_k}=\Pi_k^{\sigma/2a_k}\Psi_{A_k} Ψ A k ′ = Π k σ /2 a k Ψ A k μ k ′ \mu'_k μ k ′ e − σ E d μ k ( E ) e^{-\sigma E}d\mu_k(E) e − σ E d μ k ( E ) C k ( σ ) / C k ( 0 ) ≥ c C_k(\sigma)/C_k(0)\ge c C k ( σ ) / C k ( 0 ) ≥ c e − σ E e^{-\sigma E} e − σ E E = ∞ E=\infty E = ∞ E ≥ Λ E\ge\Lambda E ≥ Λ e − σ E ≤ e − σ Λ e^{-\sigma E}\le e^{-\sigma\Lambda} e − σ E ≤ e − σ Λ E m e − σ E ≤ ( m / e σ ) m E^m e^{-\sigma E}\le(m/e\sigma)^m E m e − σ E ≤ ( m / e σ ) m E ≥ 0 E\ge0 E ≥ 0 e − t E e^{-tE} e − tE μ k ′ \mu'_k μ k ′ ∣ e − t E − e − s E ∣ ≤ ∣ t − s ∣ E \lvert e^{-tE}-e^{-sE}\rvert\le\lvert t-s\rvert E ∣ e − tE − e − s E ∣ ≤ ∣ t − s ∣ E m = 1 m=1 m = 1 [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) { E > Λ } \{E>\Lambda\} { E > Λ } E m E^m E m
(ii) Let ν k \nu_k ν k T k T_k T k Ψ k \Psi_k Ψ k [ 0 , 1 ] [0,1] [ 0 , 1 ] n = σ / a k n=\sigma/a_k n = σ / a k λ n \lambda^n λ n
C k ( σ ) C k ( 0 ) = ∫ λ n d ν k ≥ ( ∫ λ d ν k ) n ≥ ( 1 − C a k ) n ≥ e − 2 C a k n , \frac{C_k(\sigma)}{C_k(0)}=\int\lambda^n\,d\nu_k\ge\left(\int\lambda\,d\nu_k\right)^n\ge(1-Ca_k)^n\ge e^{-2Ca_kn}, C k ( 0 ) C k ( σ ) = ∫ λ n d ν k ≥ ( ∫ λ d ν k ) n ≥ ( 1 − C a k ) n ≥ e − 2 C a k n , since log ( 1 − x ) ≥ − 2 x \log(1-x)\ge-2x log ( 1 − x ) ≥ − 2 x [ 0 , 1 2 ] [0,\frac12] [ 0 , 2 1 ]
C a k ≥ ∫ ( 1 − e − a k E ) d μ k ≥ ( 1 − e − a k Λ ) μ k ( [ Λ , ∞ ) ) , Ca_k\ge\int(1-e^{-a_kE})\,d\mu_k\ge(1-e^{-a_k\Lambda})\mu_k([\Lambda,\infty)), C a k ≥ ∫ ( 1 − e − a k E ) d μ k ≥ ( 1 − e − a k Λ ) μ k ([ Λ , ∞ )) , and 1 − e − x ≥ x / ( 1 + x ) 1-e^{-x}\ge x/(1+x) 1 − e − x ≥ x / ( 1 + x )
Part (ii) asks for an upper bound on a one-slab Dirichlet form, the opposite direction to a mass gap, which bounds such forms from below. In the box of Proposition 50, with g g g a k a_k a k Π k = A \Pi_k=A Π k = A Ψ A k = ∑ n w n Π k n u \Psi_{A_k}=\sum_n w_n\Pi_k^n u Ψ A k = ∑ n w n Π k n u w n = a k g ( a k n ) w_n=a_k g(a_kn) w n = a k g ( a k n ) u = a k F Ψ 0 u=a_kF\Psi_0 u = a k F Ψ 0 ( 1 − T k ) Ψ A k = ∑ n ( w n − w n − 1 ) Π k n u (1-T_k)\Psi_{A_k}=\sum_n(w_n-w_{n-1})\Pi_k^n u ( 1 − T k ) Ψ A k = ∑ n ( w n − w n − 1 ) Π k n u C k ≤ a k ∥ g ′ ∥ L 1 ∥ u ∥ / ∥ Ψ A k ∥ C_k\le a_k\|g'\|_{L^1}\|u\|/\|\Psi_{A_k}\| C k ≤ a k ∥ g ′ ∥ L 1 ∥ u ∥/∥ Ψ A k ∥ u u u
A one-slab bound of this kind also limits the mass that D ( n k , α ) D(n_k,\alpha) D ( n k , α ) e − m k e^{-m_k} e − m k T k T_k T k Ω k ⊥ \Omega_k^\perp Ω k ⊥ φ k ⊥ Ω k \varphi_k \perp\Omega_k φ k ⊥ Ω k ⟨ φ k , ( 1 − T k ) φ k ⟩ ≤ κ a k < 1 \langle\varphi_k,(1-T_k)\varphi_k\rangle\leq\kappa a_k < 1 ⟨ φ k , ( 1 − T k ) φ k ⟩ ≤ κ a k < 1 T k T_k T k φ k \varphi_k φ k [ 0 , e − m k ] [0,e^{-m_k}] [ 0 , e − m k ] 1 − e − m k ≤ κ a k 1-e^{-m_k} \leq\kappa a_k 1 − e − m k ≤ κ a k m k ≤ − log ( 1 − κ a k ) m_k \leq-\log(1-\kappa a_k) m k ≤ − log ( 1 − κ a k ) D ( n k , α ) D(n_k,\alpha) D ( n k , α ) n k a k ≤ ℓ n_k a_k \leq\ell n k a k ≤ ℓ Theorem 43 gives m k ≥ log ( 1 / α ) / ( 2 n k ) ≥ a k log ( 1 / α ) / ( 2 ℓ ) m_k \geq\log(1/\alpha)/(2n_k) \geq a_k\log(1/\alpha)/(2\ell) m k ≥ log ( 1/ α ) / ( 2 n k ) ≥ a k log ( 1/ α ) / ( 2 ℓ )
log ( 1 / α ) 2 ℓ ≤ − log ( 1 − κ a k ) a k = κ + O ( a k ) . \frac{\log(1/\alpha)}{2\ell} \leq\frac{-\log(1-\kappa a_k)}{a_k} = \kappa+ O(a_k). 2 ℓ log ( 1/ α ) ≤ a k − log ( 1 − κ a k ) = κ + O ( a k ) . So the physical mass m ∗ m_* m ∗ Theorem 45 is at most the constant κ \kappa κ m k → 0 m_k \to0 m k → 0
In a periodic spatial box the assumption C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0 G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) Z 4 \mathbb{Z}^4 Z 4 Z × ( Z / L k Z ) 3 \mathbb{Z}\times(\mathbb{Z}/L_k\mathbb{Z})^3 Z × ( Z / L k Z ) 3 L k ≥ 3 L_k\geq3 L k ≥ 3 ω k \omega_k ω k Theorem 48 uses only reflection positivity and the transfer matrix once the spatial translations are dropped, and parts (i)–(iv) hold in the box with D \mathcal{D} D ( t , 0 ) (t,0) ( t , 0 ) f 0 f_0 f 0 t t t
Proposition 50. In the box, let f 0 ( t , x ) = g ( t ) h ( x ) f_0(t,x)=g(t)h(x) f 0 ( t , x ) = g ( t ) h ( x ) g , h ≥ 0 g,h\geq0 g , h ≥ 0 Φ = ∑ i < j ( N − Re Tr Q i j ) \Phi=\sum_{i<j}(N-\operatorname{Re}\operatorname{Tr}Q_{ij}) Φ = ∑ i < j ( N − Re Tr Q ij ) g ( a k n ) h ( a k x ) > 0 g(a_kn)h(a_kx)>0 g ( a k n ) h ( a k x ) > 0 n ≥ 1 n\geq1 n ≥ 1 x x x C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0
Proof. Lüscher proved that the transfer matrix is strictly positive [54 ] ; we recall the argument because the proof uses it in this form.
The one-link kernel. Write χ \chi χ b = β W > 0 b=\beta_W>0 b = β W > 0
e b N Re χ = ∑ m ≥ 0 ( b / 2 N ) m m ! ( χ + χ ˉ ) m , e^{\frac{b}{N}\operatorname{Re}\chi}
=
\sum_{m\geq0}\frac{(b/2N)^m}{m!}(\chi+\bar{\chi})^m, e N b Re χ = m ≥ 0 ∑ m ! ( b /2 N ) m ( χ + χ ˉ ) m , and ( χ + χ ˉ ) m (\chi+\bar{\chi})^m ( χ + χ ˉ ) m ( ρ ⊕ ρ ˉ ) ⊗ m (\rho\oplus\bar{\rho})^{\otimes m} ( ρ ⊕ ρ ˉ ) ⊗ m σ \sigma σ q q q q ˉ \bar q q ˉ C ( G ) C(G) C ( G ) σ \sigma σ e b N Re χ e^{\frac{b}{N}\operatorname{Re}\chi} e N b Re χ L 2 ( G ) L^2(G) L 2 ( G )
The transfer matrix. In the temporal gauge the transfer matrix of the box acts on the gauge-invariant part of L 2 ( G E ) L^2(G^E) L 2 ( G E ) E E E M K M MKM M K M K K K E E E M M M [54 , 64 ] . Both commute with gauge transformations, K K K M M M ⟨ v , M K M v ⟩ = ⟨ M v , K M v ⟩ > 0 \langle v,MKMv\rangle=\langle Mv,KMv\rangle>0 ⟨ v , M K M v ⟩ = ⟨ M v , K M v ⟩ > 0 v ≠ 0 v\ne0 v = 0 K K K λ 0 \lambda_0 λ 0 ψ 0 > 0 \psi_0>0 ψ 0 > 0 A = M K M / λ 0 A=MKM/\lambda_0 A = M K M / λ 0 F F F n ≥ 0 n\geq0 n ≥ 0 Ψ F = A n F ψ 0 \Psi_F=A^nF\psi_0 Ψ F = A n F ψ 0 Ω \Omega Ω ψ 0 \psi_0 ψ 0
Conclusion. Let F = ∑ x h ( a k x ) Φ x Φ s k F=\sum_x h(a_kx)\Phi_x\Phi_{s_k} F = ∑ x h ( a k x ) Φ x Φ s k F c = F − ω k ( F ) F^c=F-\omega_k(F) F c = F − ω k ( F ) Ψ A k = a k 4 p ( A ) F c ψ 0 \Psi_{A_k}=a_k^4p(A)F^c\psi_0 Ψ A k = a k 4 p ( A ) F c ψ 0 p ( z ) = ∑ n ≥ 1 g ( a k n ) z n p(z)=\sum_{n\geq1}g(a_kn)z^n p ( z ) = ∑ n ≥ 1 g ( a k n ) z n p > 0 p>0 p > 0 ( 0 , 1 ] (0,1] ( 0 , 1 ] p ( A ) p(A) p ( A ) ∑ x h ( a k x ) Φ x \sum_x h(a_kx)\Phi_x ∑ x h ( a k x ) Φ x 1 1 1 h ( a k x ) > 0 h(a_kx)>0 h ( a k x ) > 0 Φ s k \Phi_{s_k} Φ s k F F F F F F ψ 0 > 0 \psi_0>0 ψ 0 > 0 F c ψ 0 ≠ 0 F^c\psi_0\ne0 F c ψ 0 = 0 C k ( 0 ) = ∥ Ψ A k ∥ 2 > 0 C_k(0)=\|\Psi_{A_k}\|^2>0 C k ( 0 ) = ∥ Ψ A k ∥ 2 > 0
The proposition removes the one assumption of Theorem 48 in a box; it says nothing about the ratio of part (iv), on which nontriviality depends.
The ratio of Theorem 48 (iv) can be compared with correlations of a single slice field. Suppose that f 0 ( t , x ) = g ( t ) h ( x ) f_0(t,x)=g(t)h(x) f 0 ( t , x ) = g ( t ) h ( x ) g ≥ 0 g\geq0 g ≥ 0 [ t 0 , t 1 ] [t_0,t_1] [ t 0 , t 1 ] t 0 < t 1 t_0<t_1 t 0 < t 1
U k = ∑ x ∈ Z 3 a k 3 h ( a k x ) ( Φ ( 0 , x ) ∘ Φ s k − ω k ( Φ ( 0 , x ) ∘ Φ s k ) ) U_k=\sum_{x\in\mathbb{Z}^3}a_k^3h(a_kx)\bigl(\Phi_{(0,x)}\circ\Phi_{s_k}-\omega_k(\Phi_{(0,x)}\circ\Phi_{s_k})\bigr) U k = x ∈ Z 3 ∑ a k 3 h ( a k x ) ( Φ ( 0 , x ) ∘ Φ s k − ω k ( Φ ( 0 , x ) ∘ Φ s k ) ) be the centred field of the slice at time 0 0 0 u k = Ψ U k u_k=\Psi_{U_k} u k = Ψ U k
D k ( r ) = ⟨ u k , T k r / a k u k ⟩ = ω k ( V ( U k ) T k r / a k U k ) D_k(r)=\langle u_k,T_k^r/a_ku_k\rangle=\omega_k\bigl(\mathcal{V}(U_k)T_k^{r/a_k}U_k\bigr) D k ( r ) = ⟨ u k , T k r / a k u k ⟩ = ω k ( V ( U k ) T k r / a k U k ) for r ∈ a k Z r\in a_k\mathbb{Z} r ∈ a k Z r ≥ 0 r\geq0 r ≥ 0 n 0 = ⌈ t 0 / a k ⌉ n_0=\lceil t_0/a_k\rceil n 0 = ⌈ t 0 / a k ⌉ n 1 = ⌊ t 1 / a k ⌋ n_1=\lfloor t_1/a_k\rfloor n 1 = ⌊ t 1 / a k ⌋
Proposition 51. Assume C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0 σ ≥ 0 \sigma\geq0 σ ≥ 0 σ / a k ∈ Z \sigma/a_k\in\mathbb{Z} σ / a k ∈ Z
(i) D k ( r ) > 0 D_k(r)>0 D k ( r ) > 0 r r r
D k ( 2 a k n 1 + σ ) D k ( 2 a k n 0 ) ≤ C k ( σ ) C k ( 0 ) ≤ D k ( 2 a k n 0 + σ ) D k ( 2 a k n 1 ) . \frac{D_k(2a_kn_1+\sigma)}{D_k(2a_kn_0)}
\leq
\frac{C_k(\sigma)}{C_k(0)}
\leq
\frac{D_k(2a_kn_0+\sigma)}{D_k(2a_kn_1)}. D k ( 2 a k n 0 ) D k ( 2 a k n 1 + σ ) ≤ C k ( 0 ) C k ( σ ) ≤ D k ( 2 a k n 1 ) D k ( 2 a k n 0 + σ ) . (ii) log D k \log D_k log D k log C k \log C_k log C k a k Z ≥ 0 a_k\mathbb{Z}_{\geq0} a k Z ≥ 0 D k ( r ′ ) ≥ c D k ( r ) D_k(r')\geq cD_k(r) D k ( r ′ ) ≥ c D k ( r ) r < r ′ r<r' r < r ′ c ≤ 1 c\leq1 c ≤ 1 D k ( β ) ≥ c max ( 1 , ( β − α ) / ( r ′ − r ) ) D k ( α ) D_k(\beta)\geq c^{\max(1,(\beta-\alpha)/(r'-r))}D_k(\alpha) D k ( β ) ≥ c m a x ( 1 , ( β − α ) / ( r ′ − r )) D k ( α ) r ≤ α ≤ β r\leq\alpha\leq\beta r ≤ α ≤ β C k ( σ ) ≥ c C k ( 0 ) C_k(\sigma)\geq cC_k(0) C k ( σ ) ≥ c C k ( 0 ) σ > 0 \sigma>0 σ > 0 C k ( σ ′ ) ≥ c σ ′ / σ C k ( σ ) C_k(\sigma')\geq c^{\sigma'/\sigma}C_k(\sigma) C k ( σ ′ ) ≥ c σ ′ / σ C k ( σ ) σ ′ ≥ σ \sigma'\geq\sigma σ ′ ≥ σ
(iii) Let k k k D k ( r ′ ) ≥ c D k ( r ) D_k(r')\geq cD_k(r) D k ( r ′ ) ≥ c D k ( r ) 0 < r < r ′ 0<r<r' 0 < r < r ′ r ≤ 2 t 0 r\leq2t_0 r ≤ 2 t 0 a c ∈ ( 0 , 1 ] a\,c\in(0,1] a c ∈ ( 0 , 1 ] k k k σ ≥ 0 \sigma\geq0 σ ≥ 0 k k k
C k ( σ ) ≥ c max ( 1 , ( 2 t 1 − 2 t 0 + σ ) / ( r ′ − r ) ) C k ( 0 ) . C_k(\sigma)\geq c^{\max(1,(2t_1-2t_0+\sigma)/(r'-r))}C_k(0). C k ( σ ) ≥ c m a x ( 1 , ( 2 t 1 − 2 t 0 + σ ) / ( r ′ − r )) C k ( 0 ) . Conversely, if C k ( σ ) ≥ c C k ( 0 ) C_k(\sigma)\geq cC_k(0) C k ( σ ) ≥ c C k ( 0 ) σ > 0 \sigma>0 σ > 0 k k k D k ( 2 t 0 + σ ′ ) ≥ c σ ′ / σ D k ( 2 t 1 ) D_k(2t_0+\sigma')\geq c^{\sigma'/\sigma}D_k(2t_1) D k ( 2 t 0 + σ ′ ) ≥ c σ ′ / σ D k ( 2 t 1 ) σ ′ ≥ σ \sigma'\geq\sigma σ ′ ≥ σ σ ′ > 2 ( t 1 − t 0 ) \sigma'>2(t_1-t_0) σ ′ > 2 ( t 1 − t 0 ) k k k
Proof. For a function F F F n ≥ 0 n\geq0 n ≥ 0 Ψ T k n F = T k n Ψ F \Psi_{T_k^nF}=T_k^n\Psi_F Ψ T k n F = T k n Ψ F ω k ( V ( G ) T k n F ) \omega_k(\mathcal{V}(G)T_k^nF) ω k ( V ( G ) T k n F ) Ψ G \Psi_G Ψ G ω k \omega_k ω k A k = ∑ n w n T k n A_k=\sum_n w_nT_k^n A k = ∑ n w n T k n w n = a k g ( a k n ) ≥ 0 w_n=a_kg(a_kn)\geq0 w n = a k g ( a k n ) ≥ 0
unless n 0 ≤ n ≤ n 1 n_0 \le n \le n_1 n 0 ≤ n ≤ n 1 Ψ A k = ∑ n w n T k n u k \Psi_{A_k}=\sum_n w_n T_k^n u_k Ψ A k = ∑ n w n T k n u k ν \nu ν T k T_k T k u k u_k u k [ 0 , 1 ] [0,1] [ 0 , 1 ] W ( λ ) = ∑ n w n λ n W(\lambda)=\sum_n w_n\lambda^n W ( λ ) = ∑ n w n λ n
C k ( σ ) = ∫ W ( λ ) 2 λ σ / a k d ν ( λ ) , D k ( r ) = ∫ λ r / a k d ν ( λ ) . C_k(\sigma)=\int W(\lambda)^2\lambda^{\sigma/a_k}\,d\nu(\lambda),\qquad D_k(r)=\int\lambda^{r/a_k}\,d\nu(\lambda). C k ( σ ) = ∫ W ( λ ) 2 λ σ / a k d ν ( λ ) , D k ( r ) = ∫ λ r / a k d ν ( λ ) . (i) Since t 0 > 0 t_0>0 t 0 > 0 n 0 ≥ 1 n_0\ge1 n 0 ≥ 1 W ( 0 ) = 0 W(0)=0 W ( 0 ) = 0 C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0 ν ( ( 0 , 1 ] ) > 0 \nu((0,1])>0 ν (( 0 , 1 ]) > 0 D k ( r ) > 0 D_k(r)>0 D k ( r ) > 0 0 ≤ λ ≤ 1 0\le\lambda\le1 0 ≤ λ ≤ 1 ∥ w ∥ 1 λ n 1 ≤ W ( λ ) ≤ ∥ w ∥ 1 λ n 0 \lVert w\rVert_1\lambda^{n_1}\le W(\lambda)\le\lVert w\rVert_1\lambda^{n_0} ∥ w ∥ 1 λ n 1 ≤ W ( λ ) ≤ ∥ w ∥ 1 λ n 0 C k ( σ ) C_k(\sigma) C k ( σ ) C k ( 0 ) C_k(0) C k ( 0 )
(ii) Convexity of j ↦ log ∫ λ j d ρ j\mapsto\log\int\lambda^j\,d\rho j ↦ log ∫ λ j d ρ ρ \rho ρ [ 0 , 1 ] [0,1] [ 0 , 1 ] j j j λ ≤ 1 \lambda\le1 λ ≤ 1 ρ = ν \rho=\nu ρ = ν ρ = W 2 ν \rho=W^2\nu ρ = W 2 ν β ≤ r ′ \beta\le r' β ≤ r ′ D k ( β ) ≥ D k ( r ′ ) ≥ c D k ( r ) ≥ c D k ( α ) D_k(\beta)\ge D_k(r')\ge cD_k(r)\ge cD_k(\alpha) D k ( β ) ≥ D k ( r ′ ) ≥ c D k ( r ) ≥ c D k ( α ) β > r ′ \beta>r' β > r ′ log D k \log D_k log D k
log D k ( β ) − log D k ( α ) β − α ≥ log D k ( r ′ ) − log D k ( r ) r ′ − r ≥ log c r ′ − r . \frac{\log D_k(\beta)-\log D_k(\alpha)}{\beta-\alpha}\ge\frac{\log D_k(r')-\log D_k(r)}{r'-r}\ge\frac{\log c}{r'-r}. β − α log D k ( β ) − log D k ( α ) ≥ r ′ − r log D k ( r ′ ) − log D k ( r ) ≥ r ′ − r log c . The statement on C k C_k C k [ 0 , σ ] [0,\sigma] [ 0 , σ ] [ 0 , σ ′ ] [0,\sigma'] [ 0 , σ ′ ]
(iii) For large k k k 2 t 0 , 2 t 1 2t_0,2t_1 2 t 0 , 2 t 1 a k a_k a k 2 a k n 0 ≥ 2 t 0 2a_kn_0\ge2t_0 2 a k n 0 ≥ 2 t 0 2 a k n 1 ≤ 2 t 1 2a_kn_1\le2t_1 2 a k n 1 ≤ 2 t 1 D k D_k D k
D k ( 2 t 1 + σ ) D k ( 2 t 0 ) ≤ C k ( σ ) C k ( 0 ) ≤ D k ( 2 t 0 + σ ) D k ( 2 t 1 ) . \frac{D_k(2t_1+\sigma)}{D_k(2t_0)}
\le\frac{C_k(\sigma)}{C_k(0)}
\le\frac{D_k(2t_0+\sigma)}{D_k(2t_1)}. D k ( 2 t 0 ) D k ( 2 t 1 + σ ) ≤ C k ( 0 ) C k ( σ ) ≤ D k ( 2 t 1 ) D k ( 2 t 0 + σ ) . The first claim is the left inequality with (ii) for α = 2 t 0 ≥ r \alpha=2t_0\ge r α = 2 t 0 ≥ r β = 2 t 1 + σ \beta=2t_1+\sigma β = 2 t 1 + σ C k ( σ ′ ) ≥ c ′ / σ C k ( 0 ) C_k(\sigma')\ge c^{\prime/\sigma}C_k(0) C k ( σ ′ ) ≥ c ′/ σ C k ( 0 ) σ ′ \sigma' σ ′ σ ′ > 2 ( t 1 − t 0 ) \sigma'>2(t_1-t_0) σ ′ > 2 ( t 1 − t 0 )
So for the packet g ⊗ h g\otimes h g ⊗ h Theorem 48 (iv) follows from an inequality between two correlations of one slice field,
lim inf k D k ( r ′ ) D k ( r ) > 0 \liminf_k\frac{D_k(r')}{D_k(r)}>0 k lim inf D k ( r ) D k ( r ′ ) > 0 for one pair 0 < r < r ′ 0<r<r' 0 < r < r ′ r ≤ 2 t 0 r\le2t_0 r ≤ 2 t 0 g ′ ⊗ h g'\otimes h g ′ ⊗ h g ′ g' g ′ [ t 1 , ∞ ) [t_1,\infty) [ t 1 , ∞ ) r ≤ 2 t 0 r\le2t_0 r ≤ 2 t 0
D k ( r ′ ) / D k ( r ) = ( 1 − 4 − k ) 2 − k r ′ + 4 − k 2 − r D_k(r')/D_k(r)=(1-4^{-k})2^{-k r'}+4^{-k2^{-r}} D k ( r ′ ) / D k ( r ) = ( 1 − 4 − k ) 2 − k r ′ + 4 − k 2 − r shows, with D k ( 4 ) / D k ( 3 ) → 1 2 D_k(4)/D_k(3)\to\frac12 D k ( 4 ) / D k ( 3 ) → 2 1 D k ( 2 ) / D k ( 1 ) → 0 D_k(2)/D_k(1)\to0 D k ( 2 ) / D k ( 1 ) → 0 u u u Theorem 45 , is an upper bound of the form e − m ( r ′ − r ) / a k e^{-m(r'-r)/a_k} e − m ( r ′ − r ) / a k
Remark 52. Let D k D_k D k a k Z 4 a_k\mathbb{Z}^4 a k Z 4 Φ \Phi Φ h ∈ C c ∞ ( R 3 ) h\in C^\infty_c(\mathbb{R}^3) h ∈ C c ∞ ( R 3 ) t f > 0 t_f>0 t f > 0
p ^ 2 = ∑ i ( 2 a k − 1 sin ( a k p i / 2 ) ) 2 , cosh ( a k ω ) = 1 + a k 2 p ^ 2 / 2 \widehat p^2=\sum_i(2a_k^{-1}\sin(a_kp_i/2))^2,\qquad\cosh(a_k\omega)=1+a_k^2\widehat p^2/2 p 2 = i ∑ ( 2 a k − 1 sin ( a k p i /2 ) ) 2 , cosh ( a k ω ) = 1 + a k 2 p 2 /2 and h ^ k ( p ) = ∑ x a k 3 h ( a k x ) e − i a k p ⋅ x \widehat h_k(p)=\sum_xa_k^3h(a_kx)e^{-ia_kp\cdot x} h k ( p ) = ∑ x a k 3 h ( a k x ) e − i a k p ⋅ x
D k ( r ) = ∫ [ − π / a k , π / a k ] 3 ∣ h ^ k ( p ) ∣ 2 e − 2 t f p ^ 2 e − r ω 2 a k − 1 sinh ( a k ω ) d 3 p ( 2 π ) 3 . D_k(r)=\int_{[-\pi/a_k,\pi/a_k]^3}|\widehat h_k(p)|^2e^{-2t_f\widehat p^2}\frac{e^{-r\omega}}{2a_k^{-1}\sinh(a_k\omega)}\frac{d^3p}{(2\pi)^3}. D k ( r ) = ∫ [ − π / a k , π / a k ] 3 ∣ h k ( p ) ∣ 2 e − 2 t f p 2 2 a k − 1 sinh ( a k ω ) e − r ω ( 2 π ) 3 d 3 p . On the Brillouin zone p ^ ≥ 2 ∣ p ∣ / π \widehat p\ge2|p|/\pi p ≥ 2∣ p ∣/ π a k 2 p ^ 2 / 2 < 6 a_k^2\widehat p^2/2<6 a k 2 p 2 /2 < 6 arccosh ( 1 + y ) / 2 y \operatorname{arccosh}(1+y)/\sqrt{2y} arccosh ( 1 + y ) / 2 y ( 0 , 6 ] (0,6] ( 0 , 6 ] a k ω ≥ κ a k p ^ a_k\omega\ge\kappa a_k\widehat p a k ω ≥ κ a k p κ = arccosh 7 / 12 > 0.76 \kappa=\operatorname{arccosh}7/\sqrt{12}>0.76 κ = arccosh 7/ 12 > 0.76 a k − 1 sinh ( a k ω ) ≥ ω a_k^{-1}\sinh(a_k\omega)\ge\omega a k − 1 sinh ( a k ω ) ≥ ω ∣ h ^ k ∣ |\widehat h_k| ∣ h k ∣ k k k e − 8 t f ∣ p ∣ 2 / π 2 / ∣ p ∣ e^{-8t_f|p|^2/\pi^2}/|p| e − 8 t f ∣ p ∣ 2 / π 2 /∣ p ∣ R 3 \mathbb{R}^3 R 3 ∣ h ^ ( p ) ∣ 2 e − 2 t f ∣ p ∣ 2 e − r ∣ p ∣ / ( 2 ∣ p ∣ ) |\widehat h(p)|^2e^{-2t_f|p|^2}e^{-r|p|}/(2|p|) ∣ h ( p ) ∣ 2 e − 2 t f ∣ p ∣ 2 e − r ∣ p ∣ / ( 2∣ p ∣ ) D k ( r ′ ) / D k ( r ) D_k(r')/D_k(r) D k ( r ′ ) / D k ( r ) h = e − ∣ x ∣ 2 h=e^{-|x|^2} h = e − ∣ x ∣ 2 t f = 0.05 t_f=0.05 t f = 0.05 r = 1 2 r=\frac12 r = 2 1 r ′ = 1 r'=1 r ′ = 1 0.6337 0.6337 0.6337
Remark 53. The inequality also holds along the right tuning in two-dimensional S U ( N ) \mathrm{SU}(N) SU ( N ) a k Z × a k Z L k a_k\mathbb{Z}\times a_k\mathbb{Z}_{L_k} a k Z × a k Z L k a k L k = ℓ a_kL_k=\ell a k L k = ℓ ∏ p exp ( β k N Re Tr U p ) \prod_p\exp(\frac{\beta_k}{N}\operatorname{Re}\operatorname{Tr}U_p) ∏ p exp ( N β k Re Tr U p ) β k = 2 N / ( g 2 a k 2 ) \beta_k=2N/(g^2a_k^2) β k = 2 N / ( g 2 a k 2 ) V V V w β w_\beta w β e β N Re Tr U e^{\frac{\beta}{N}\operatorname{Re}\operatorname{Tr}U} e N β Re Tr U λ \lambda λ d λ d_\lambda d λ χ λ \chi_\lambda χ λ c ^ λ ( β ) = d λ − 1 ∫ χ λ w β d U \widehat c_\lambda(\beta)=d_\lambda^{-1}\int\chi_\lambda w_\beta\,dU c λ ( β ) = d λ − 1 ∫ χ λ w β d U W x = V x ′ V x − 1 W_x=V_x'V_x^{-1} W x = V x ′ V x − 1 W x W_x W x w β k w_{\beta_k} w β k
V 1 ′ ⋯ V L ′ = W 1 ( V 1 W 2 V 1 − 1 ) ⋯ ( V 1 ⋯ V L − 1 W L ( V 1 ⋯ V L − 1 ) − 1 ) V 1 ⋯ V L , V_1'\cdots V_L'=W_1(V_1W_2V_1^{-1})\cdots(V_1\cdots V_{L-1}W_L(V_1\cdots V_{L-1})^{-1})V_1\cdots V_L, V 1 ′ ⋯ V L ′ = W 1 ( V 1 W 2 V 1 − 1 ) ⋯ ( V 1 ⋯ V L − 1 W L ( V 1 ⋯ V L − 1 ) − 1 ) V 1 ⋯ V L , so the new holonomy is Q V QV Q V Q Q Q L k L_k L k w β k w_{\beta_k} w β k V V V Lemma 120 the transfer matrix acts on class functions by χ λ ↦ c ^ λ ( β k ) L k χ λ \chi_\lambda\mapsto\widehat c_\lambda(\beta_k)^{L_k}\chi_\lambda χ λ ↦ c λ ( β k ) L k χ λ w β w_\beta w β U ↦ U ‾ U\mapsto\overline U U ↦ U β 2 N ( χ F + χ F ‾ ) \frac{\beta}{2N}(\chi_F+\overline{\chi_F}) 2 N β ( χ F + χ F ) 1 1 1 λ ≠ 0 \lambda\ne0 λ = 0 f f f ∫ f d U = 0 \int f\,dU=0 ∫ f d U = 0 ∣ f ^ λ ∣ |\widehat f_\lambda| ∣ f λ ∣
D k ( r ) = ∑ λ ≠ 0 ∣ f ^ λ ∣ 2 e − r E λ , k , E λ , k = − ℓ a k 2 log c ^ λ ( β k ) . D_k(r)=\sum_{\lambda\ne0}|\widehat f_\lambda|^2e^{-rE_{\lambda,k}},\qquad E_{\lambda,k}=-\frac{\ell}{a_k^2}\log\widehat c_\lambda(\beta_k). D k ( r ) = λ = 0 ∑ ∣ f λ ∣ 2 e − r E λ , k , E λ , k = − a k 2 ℓ log c λ ( β k ) . Laplace’s method[16 ] at the unique maximum U = 1 U=1 U = 1 Re Tr U \operatorname{Re}\operatorname{Tr}U Re Tr U U = e X U=e^X U = e X Re Tr e X = N + 1 2 Tr X 2 + O ( ∣ X ∣ 4 ) \operatorname{Re}\operatorname{Tr}e^X=N+\frac12\operatorname{Tr}X^2+O(|X|^4) Re Tr e X = N + 2 1 Tr X 2 + O ( ∣ X ∣ 4 ) Re Tr X 3 = 0 \operatorname{Re}\operatorname{Tr}X^3=0 Re Tr X 3 = 0 Tr ρ λ ( X ) = 0 \operatorname{Tr}\rho_\lambda(X)=0 Tr ρ λ ( X ) = 0 ∑ a ρ λ ( T a ) 2 = − 2 C 2 ( λ ) \sum_a\rho_\lambda(T_a)^2=-2C_2(\lambda) ∑ a ρ λ ( T a ) 2 = − 2 C 2 ( λ ) s u ( N ) \mathfrak{su}(N) su ( N ) − Tr X 2 -\operatorname{Tr}X^2 − Tr X 2
c ^ λ ( β ) = 1 − N C 2 ( λ ) β + O ( β − 2 ) , \widehat c_\lambda(\beta)=1-\frac{NC_2(\lambda)}{\beta}+O(\beta^{-2}), c λ ( β ) = 1 − β N C 2 ( λ ) + O ( β − 2 ) , with the Casimir normalised by C 2 = ( N 2 − 1 ) / ( 2 N ) C_2=(N^2-1)/(2N) C 2 = ( N 2 − 1 ) / ( 2 N ) Lemma 121 ). Hence E λ , k → g 2 ℓ C 2 ( λ ) / 2 E_{\lambda,k}\to g^2\ell C_2(\lambda)/2 E λ , k → g 2 ℓ C 2 ( λ ) /2 D k ( r ) ≤ ∥ f ∥ 2 D_k(r)\le\lVert f\rVert^2 D k ( r ) ≤ ∥ f ∥ 2
lim inf k D k ( r ′ ) D k ( r ) ≥ ∣ f ^ μ ∣ 2 ∥ f ∥ 2 e − r ′ g 2 ℓ C 2 ( μ ) / 2 > 0. \liminf_k\frac{D_k(r')}{D_k(r)}\ge\frac{|\widehat f_\mu|^2}{\lVert f\rVert^2}e^{-r'g^2\ell C_2(\mu)/2}>0. k lim inf D k ( r ) D k ( r ′ ) ≥ ∥ f ∥ 2 ∣ f μ ∣ 2 e − r ′ g 2 ℓ C 2 ( μ ) /2 > 0. for every μ ≠ 0 \mu\ne0 μ = 0 f ^ μ ≠ 0 \hat{f}_{\mu} \ne0 f ^ μ = 0 f = ReTr V f = \operatorname{ReTr}V f = ReTr V e − ( r ′ − r ) E F , k → e − ( r ′ − r ) g 2 ℓ 2 ( N 2 − 1 ) / ( 4 N ) e^{-(r'-r)E_{F,k}} \to e^{-(r'-r)g^2\ell^2(N^2-1)/(4N)} e − ( r ′ − r ) E F , k → e − ( r ′ − r ) g 2 ℓ 2 ( N 2 − 1 ) / ( 4 N ) [32 , 40 , 52 ] , written as the inequality. In two dimensions g 2 g^2 g 2 M k M_k M k m k = a k − 1 arccosh ( 1 + a k 2 M k 2 / 2 ) m_k = a_k^{-1}\operatorname{arccosh}(1+a_k^2M_k^2/2) m k = a k − 1 arccosh ( 1 + a k 2 M k 2 /2 ) D k ( r ′ ) ≤ e − m k ( r ′ − r ) D k ( r ) D_k(r') \le e^{-m_k(r'-r)}D_k(r) D k ( r ′ ) ≤ e − m k ( r ′ − r ) D k ( r ) D k ( r ) D_k(r) D k ( r ) a k M k a_kM_k a k M k
A route to the slice ratio in four dimensions replaces the packet by its conditional expectations given averages of the field over blocks of increasing size and controls the error of each step. The free field shows in which norm such errors can be small. In the ordinary L 2 L^2 L 2
Proposition 54. Let d ≥ 2 d \ge2 d ≥ 2 Λ \Lambda Λ Z d \mathbb{Z}^d Z d μ > 0 \mu> 0 μ > 0 ϕ \phi ϕ Λ \Lambda Λ exp ( − 1 2 ∑ x , ν ( ∇ ν ϕ ( x ) ) 2 − μ 2 2 ∑ x ϕ ( x ) 2 ) \exp\left(-\frac{1}{2}\sum_{x,\nu}(\nabla_\nu\phi(x))^2-\frac{\mu^2}{2}\sum_x\phi(x)^2\right) exp ( − 2 1 ∑ x , ν ( ∇ ν ϕ ( x ) ) 2 − 2 μ 2 ∑ x ϕ ( x ) 2 ) ∇ ν ϕ ( x ) = ϕ ( x + e ν ) − ϕ ( x ) \nabla_\nu\phi(x)=\phi(x+e_\nu)-\phi(x) ∇ ν ϕ ( x ) = ϕ ( x + e ν ) − ϕ ( x ) Z \mathcal{Z} Z B = 2 z + { 0 , 1 } d B=2z+\{0,1\}^d B = 2 z + { 0 , 1 } d B \mathcal{B} B σ \sigma σ ϕ \phi ϕ Z \mathcal{Z} Z f ≥ 0 f\ge0 f ≥ 0 Λ \Lambda Λ
F = ∑ x ∈ Λ f ( x ) ∑ ν = 1 d : ( ∇ ν ϕ ( x ) ) 2 : ; F=\sum_{x\in\Lambda}f(x)\sum_{\nu=1}^{d}:(\nabla_\nu\phi(x))^2:; F = x ∈ Λ ∑ f ( x ) ν = 1 ∑ d : ( ∇ ν ϕ ( x ) ) 2 : ; where : X 2 : = X 2 − E X 2 :X^2:=X^2-\mathbb{E}X^2 : X 2 := X 2 − E X 2 B = 2 z + { 0 , 1 } d B=2z+\{0,1\}^d B = 2 z + { 0 , 1 } d x B = 2 z x_B=2z x B = 2 z y B = 2 z + e 1 + e 2 y_B=2z+e_1+e_2 y B = 2 z + e 1 + e 2
α B = 1 2 ∑ w ∈ { x B , y B } ( d f ( w ) + ∑ ν = 1 d f ( w − e ν ) ) , \alpha_B=\frac{1}{2}\sum_{w\in\{x_B,y_B\}}\left(df(w)+\sum_{\nu=1}^{d}f(w-e_\nu)\right), α B = 2 1 w ∈ { x B , y B } ∑ ( df ( w ) + ν = 1 ∑ d f ( w − e ν ) ) , and σ 2 = 2 / ( 2 d + μ 2 ) \sigma^2=2/(2d+\mu^2) σ 2 = 2/ ( 2 d + μ 2 ) E Var ( F ∣ B ) ≥ 2 σ 4 ∑ B ∈ Z α B 2 \mathbb{E}\operatorname{Var}(F\mid\mathcal{B})\ge2\sigma^4\sum_{B\in\mathcal{Z}}\alpha_B^2 E Var ( F ∣ B ) ≥ 2 σ 4 ∑ B ∈ Z α B 2 Var F ≤ 2 d 2 ∑ x f ( x ) 2 \operatorname{Var}F\le2d^2\sum_x f(x)^2 Var F ≤ 2 d 2 ∑ x f ( x ) 2 g ≥ 0 g\ge0 g ≥ 0 R d \mathbb{R}^d R d ∫ g 2 > 0 \int g^2>0 ∫ g 2 > 0 n ≥ 1 n\ge1 n ≥ 1 Λ n \Lambda_n Λ n n supp g n\operatorname{supp}g n supp g f n ( x ) = g ( x / n ) f_n(x)=g(x/n) f n ( x ) = g ( x / n ) Z n \mathcal{Z}_n Z n n supp g n\operatorname{supp}g n supp g μ = μ n → 0 \mu=\mu_n\to0 μ = μ n → 0 F n F_n F n f n f_n f n
lim inf n → ∞ E Var ( F n ∣ B n ) n d ∫ g 2 ≥ 2 1 − d , lim sup n → ∞ Var F n n d ∫ g 2 ≤ 2 d 2 , \liminf_{n\to\infty}\frac{\mathbb{E}\operatorname{Var}(F_n\mid\mathcal{B}_n)}{n^d\int g^2}\ge2^{1-d},\qquad
\limsup_{n\to\infty}\frac{\operatorname{Var}F_n}{n^d\int g^2}\le2d^2, n → ∞ lim inf n d ∫ g 2 E Var ( F n ∣ B n ) ≥ 2 1 − d , n → ∞ lim sup n d ∫ g 2 Var F n ≤ 2 d 2 , so E Var ( F n ∣ B n ) ≥ ( 2 − d d − 2 − o ( 1 ) ) Var F n \mathbb{E}\operatorname{Var}(F_n\mid\mathcal{B}_n)\ge(2^{-d}d^{-2}-o(1))\operatorname{Var}F_n E Var ( F n ∣ B n ) ≥ ( 2 − d d − 2 − o ( 1 )) Var F n
Proof. (i) No two of the sites x B , y B x_B,y_B x B , y B B ∈ Z B\in\mathcal{Z} B ∈ Z H ⊇ B \mathcal{H}\supseteq\mathcal{B} H ⊇ B B \mathcal{B} B ϕ ( w ) \phi(w) ϕ ( w ) w w w { x B , y B : B ∈ Z } \{x_B,y_B:B\in\mathcal{Z}\} { x B , y B : B ∈ Z } Var ( F ∣ B ) = E [ Var ( F ∣ H ) ∣ B ] + Var ( E [ F ∣ H ] ∣ B ) \operatorname{Var}(F\mid\mathcal{B})=\mathbb{E}[\operatorname{Var}(F\mid\mathcal{H})\mid\mathcal{B}]+\operatorname{Var}(\mathbb{E}[F\mid\mathcal{H}]\mid\mathcal{B}) Var ( F ∣ B ) = E [ Var ( F ∣ H ) ∣ B ] + Var ( E [ F ∣ H ] ∣ B ) E Var ( F ∣ B ) ≥ E Var ( F ∣ H ) \mathbb{E}\operatorname{Var}(F\mid\mathcal{B})\ge\mathbb{E}\operatorname{Var}(F\mid\mathcal{H}) E Var ( F ∣ B ) ≥ E Var ( F ∣ H ) w w w ( d + μ 2 2 ) ϕ ( w ) 2 (d+\frac{\mu^2}{2})\phi(w)^2 ( d + 2 μ 2 ) ϕ ( w ) 2 − ϕ ( w ) ϕ ( w ′ ) -\phi(w)\phi(w') − ϕ ( w ) ϕ ( w ′ ) 2 d 2d 2 d w ′ w' w ′ H \mathcal{H} H s B = ϕ ( x B ) + ϕ ( y B ) s_B=\phi(x_B)+\phi(y_B) s B = ϕ ( x B ) + ϕ ( y B ) u B = ϕ ( x B ) − ϕ ( y B ) u_B=\phi(x_B)-\phi(y_B) u B = ϕ ( x B ) − ϕ ( y B ) ϕ ( x B ) 2 + ϕ ( y B ) 2 = 1 2 ( s B 2 + u B 2 ) \phi(x_B)^2+\phi(y_B)^2=\frac{1}{2}(s_B^2+u_B^2) ϕ ( x B ) 2 + ϕ ( y B ) 2 = 2 1 ( s B 2 + u B 2 ) H \mathcal{H} H ( u B ) B ∈ Z (u_B)_{B\in\mathcal{Z}} ( u B ) B ∈ Z ∏ B exp ( − 1 4 ( 2 d + μ 2 ) u B 2 + c B u B ) \prod_B\exp\left(-\frac{1}{4}(2d+\mu^2)u_B^2+c_Bu_B\right) ∏ B exp ( − 4 1 ( 2 d + μ 2 ) u B 2 + c B u B ) H \mathcal{H} H c B c_B c B u B u_B u B σ 2 \sigma^2 σ 2 ( ∇ ν ϕ ( w ) ) 2 (\nabla_\nu\phi(w))^2 ( ∇ ν ϕ ( w ) ) 2 x B x_B x B ϕ ( x B ) = 1 2 ( s B + u B ) \phi(x_B)=\frac{1}{2}(s_B+u_B) ϕ ( x B ) = 2 1 ( s B + u B ) 1 4 u B 2 \frac{1}{4}u_B^2 4 1 u B 2 u B u_B u B H \mathcal{H} H y B y_B y B − u B -u_B − u B x B x_B x B w = x B w=x_B w = x B w = x B − e ν w=x_B-e_\nu w = x B − e ν d d d
F = ∑ B ∈ Z ( α B u B 2 + β B u B ) + F 0 F=\sum_{B\in\mathcal{Z}}(\alpha_Bu_B^2+\beta_Bu_B)+F_0 F = B ∈ Z ∑ ( α B u B 2 + β B u B ) + F 0 with H \mathcal{H} H β B \beta_B β B F 0 F_0 F 0 u B = m B + v B u_B=m_B+v_B u B = m B + v B m B m_B m B α B u B 2 + β B u B = α B v B 2 + ( 2 α B m B + β B ) v B + const \alpha_Bu_B^2+\beta_Bu_B=\alpha_Bv_B^2+(2\alpha_Bm_B+\beta_B)v_B+\operatorname{const} α B u B 2 + β B u B = α B v B 2 + ( 2 α B m B + β B ) v B + const v v v σ 2 \sigma^2 σ 2 Var ( v 2 ) = 2 σ 4 \operatorname{Var}(v^2)=2\sigma^4 Var ( v 2 ) = 2 σ 4 γ v = γ 2 σ 2 \gamma v= \gamma^2\sigma^2 γ v = γ 2 σ 2 Var ( F ∣ H ) ≥ 2 σ 4 ∑ B α B 2 \operatorname{Var}(F\mid\mathcal{H})\ge2\sigma^4\sum_B\alpha_B^2 Var ( F ∣ H ) ≥ 2 σ 4 ∑ B α B 2
(ii) For centred jointly Gaussian X , Y X,Y X , Y Cov ( X 2 : Y 2 : ) = 2 Cov ( X , Y ) 2 \operatorname{Cov}(X^2:Y^2:)=2\operatorname{Cov}(X,Y)^2 Cov ( X 2 : Y 2 : ) = 2 Cov ( X , Y ) 2 G ν ρ ( x , y ) = E [ ∇ ν ϕ ( x ) ∇ ρ ϕ ( y ) ] G_{\nu\rho}(x,y)=\mathbb{E}[\nabla_\nu\phi(x)\nabla_\rho\phi(y)] G ν ρ ( x , y ) = E [ ∇ ν ϕ ( x ) ∇ ρ ϕ ( y )]
Var F = 2 ∑ x , y f ( x ) f ( y ) ∑ ν , ρ G ν ρ ( x , y ) 2 ≤ ∑ x , y ( f ( x ) 2 + f ( y ) 2 ) ∑ ν , ρ G ν ρ ( x , y ) 2 . \begin{aligned}
\operatorname{Var}F=2\sum_{x,y}f(x)f(y)\sum_{\nu,\rho}G_{\nu\rho}(x,y)^2 \\
&\le\sum_{x,y}(f(x)^2+f(y)^2)\sum_{\nu,\rho}G_{\nu\rho}(x,y)^2.
\end{aligned} Var F = 2 x , y ∑ f ( x ) f ( y ) ν , ρ ∑ G ν ρ ( x , y ) 2 ≤ x , y ∑ ( f ( x ) 2 + f ( y ) 2 ) ν , ρ ∑ G ν ρ ( x , y ) 2 . On the dual torus Λ ∗ \Lambda^* Λ ∗ p ^ 2 = ∑ ν ∣ e i p ν − 1 ∣ 2 \hat p^2=\sum_\nu|e^{ip_\nu}-1|^2 p ^ 2 = ∑ ν ∣ e i p ν − 1 ∣ 2
G ν ρ ( x , y ) = 1 ∣ Λ ∣ ∑ p ∈ Λ ∗ ( e i p ν − 1 ) ( e − i p ρ − 1 ) p ^ 2 + μ 2 e i p ⋅ ( x − y ) , G_{\nu\rho}(x,y)=\frac{1}{|\Lambda|}\sum_{p\in\Lambda^*}\frac{(e^{ip_\nu}-1)(e^{-ip_\rho}-1)}{\hat p^2+\mu^2}e^{ip\cdot(x-y)}, G ν ρ ( x , y ) = ∣Λ∣ 1 p ∈ Λ ∗ ∑ p ^ 2 + μ 2 ( e i p ν − 1 ) ( e − i p ρ − 1 ) e i p ⋅ ( x − y ) , and by Parseval ∑ y G ν ρ ( x , y ) 2 = ∣ Λ ∣ − 1 ∑ p ∣ e i p ν − 1 ∣ 2 ∣ e i p ρ − 1 ∣ 2 ( p ^ 2 + μ 2 ) − 2 ≤ 1 \sum_yG_{\nu\rho}(x,y)^2=|\Lambda|^{-1}\sum_p|e^{ip_\nu}-1|^2|e^{ip_\rho}-1|^2(\hat p^2+\mu^2)^{-2}\le1 ∑ y G ν ρ ( x , y ) 2 = ∣Λ ∣ − 1 ∑ p ∣ e i p ν − 1 ∣ 2 ∣ e i p ρ − 1 ∣ 2 ( p ^ 2 + μ 2 ) − 2 ≤ 1 ∣ e i p ν − 1 ∣ 2 |e^{ip_\nu}-1|^2 ∣ e i p ν − 1 ∣ 2 p ^ 2 \hat p^2 p ^ 2 x x x
(iii) If ∣ p ∣ = K |p|=K ∣ p ∣ = K α B \alpha_B α B 3 3 3 2 z 2z 2 z ∣ α B − d g ( 2 z / n ) ∣ ≤ 3 d K / n |\alpha_B-dg(2z/n)|\le3dK/n ∣ α B − d g ( 2 z / n ) ∣ ≤ 3 d K / n ∑ B ∈ Z n α B 2 ≥ ∑ z ( d g ( 2 z / n ) − 3 d K / n ) 2 \sum_{B\in\mathcal{Z}_n}\alpha_B^2\ge\sum_z(dg(2z/n)-3dK/n)^2 ∑ B ∈ Z n α B 2 ≥ ∑ z ( d g ( 2 z / n ) − 3 d K / n ) 2 z z z 2 z ∈ n supp g 2z\in n\operatorname{supp}g 2 z ∈ n supp g d 2 ( n / 2 ) d ∫ g 2 ( 1 + o ( 1 ) ) d^2(n/2)^d\int g^2(1+o(1)) d 2 ( n /2 ) d ∫ g 2 ( 1 + o ( 1 )) ( 2 / n ) Z d (2/n)\mathbb{Z}^d ( 2/ n ) Z d σ 2 = 4 / ( 2 d + μ n 2 ) → 2 / d \sigma^2=4/(2d+\mu_n^2)\to2/d^{} σ 2 = 4/ ( 2 d + μ n 2 ) → 2/ d ∑ x g ( x / n ) 2 = n d ∫ g 2 ( 1 + o ( 1 ) ) \sum_xg(x/n)^2=n^d\int g^2(1+o(1)) ∑ x g ( x / n ) 2 = n d ∫ g 2 ( 1 + o ( 1 )) □ \square □
For d = 4 d=4 d = 4 Var F n \operatorname{Var}F_n Var F n ( ℓ / a ) 4 (\ell/a)^4 ( ℓ / a ) 4 ℓ \ell ℓ 1 − 2 − d d − 2 + o ( 1 ) 1-2^{-d}d^{-2}+o(1) 1 − 2 − d d − 2 + o ( 1 ) C k ( 0 ) = ω k ( ϑ ( A ˉ k ) A k ) C_k(0)=\omega_k(\vartheta(\bar A_k)A_k) C k ( 0 ) = ω k ( ϑ ( A ˉ k ) A k ) [48 ] , involves the covariance only between the support of ϑ f 0 \vartheta f_0 ϑ f 0 f 0 f_0 f 0 Figure 12 shows the blocks between the reflection plane and the packet.
Figure 12. Block averages between the reflection plane t = 0 t=0 t = 0 F F F d d d d / 4 d/4 d /4 d / 2 d/2 d /2 d d d
Figure 10. Two-dimensional SU(N) (Remark 53). (a) The lattice a k Z × a k Z L k a_k\mathbb{Z}\times a_k\mathbb{Z}_{L_k} a k Z × a k Z L k V V V V ′ = Q V V'=QV V ′ = Q V W x W_x W x E j , k = − ( ℓ / a k 2 ) log c ^ j ( β k ) E_{j,k}=-(\ell/a_k^2)\log\hat{c}_j(\beta_k) E j , k = − ( ℓ / a k 2 ) log c ^ j ( β k ) g 2 = ℓ = 1 g^2=\ell=1 g 2 = ℓ = 1 c ^ j ( β ) = I 2 j + 1 ( β ) / I 1 ( β ) \hat{c}_j(\beta)=I_{2j+1}(\beta)/I_1(\beta) c ^ j ( β ) = I 2 j + 1 ( β ) / I 1 ( β ) j ( j + 1 ) / 2 j(j+1)/2 j ( j + 1 ) /2
Figure 11. The ratio D k ( r ) / D k ( 1 2 ) D_k(r)/D_k(\frac{1}{2}) D k ( r ) / D k ( 2 1 ) r r r a k a_k a k f = Re Tr V f=\operatorname{Re}\operatorname{Tr}V f = Re Tr V g 2 = ℓ = 1 g^2=\ell=1 g 2 = ℓ = 1 e − ( r − 1 / 2 ) E 1 / 2 , k e^{-(r-1/2)E_{1/2,k}} e − ( r − 1/2 ) E 1/2 , k a k → 0 a_k\to0 a k → 0 e − ( r − 1 / 2 ) m k e^{-(r-1/2)m_k} e − ( r − 1/2 ) m k m k m_k m k 1 2 \frac{1}{2} 2 1 r > 1 2 r>\frac{1}{2} r > 2 1
Higher correlations
Theorem 48 controls two renormalised fields at a time, and part (i) of it gives nothing for more: Z k Z_k Z k Φ x ∘ Φ s k \Phi_x\circ\Phi_{s_k} Φ x ∘ Φ s k Φ \Phi Φ φ k ( x ) = Φ x ∘ Φ s k \varphi_k(x)=\Phi_x\circ\Phi_{s_k} φ k ( x ) = Φ x ∘ Φ s k ω k T \omega_k^T ω k T ω k \omega_k ω k
C c ( R 4 ) C_c(\mathbb{R}^4) C c ( R 4 )
A k ( f ) = ∑ x ∈ Z 4 a k 4 f ( a k x ) ( φ k ( x ) − ω k ( φ k ( x ) ) ) , A_k(f)=\sum_{x\in\mathbb{Z}^4}a_k^4f(a_kx)\left(\varphi_k(x)-\omega_k(\varphi_k(x))\right), A k ( f ) = x ∈ Z 4 ∑ a k 4 f ( a k x ) ( φ k ( x ) − ω k ( φ k ( x )) ) , so that A k = A k ( f 0 ) A_k=A_k(f_0) A k = A k ( f 0 ) A ˉ k ( f ) = A k ( f ˉ ) \bar A_k(f)=A_k(\bar f) A ˉ k ( f ) = A k ( f ˉ ) w k : Z 4 → [ 0 , ∞ ) w_k:\mathbb{Z}^4\to[0,\infty) w k : Z 4 → [ 0 , ∞ ) W k = ∑ x w k ( x ) < ∞ W_k=\sum_xw_k(x)<\infty W k = ∑ x w k ( x ) < ∞ r k > 0 r_k>0 r k > 0
(T1) Tree bounds. For every n ≥ 2 n\ge2 n ≥ 2 K n K_n K n k k k
∣ ω k T ( φ k ( x 1 ) ; … ; φ k ( x n ) ) ∣ ≤ K n r k n ∑ T ∏ { i , j } ∈ T w k ( x i − x j ) \left|\omega_k^{\mathrm{T}}\left(\varphi_k(x_1);\ldots;\varphi_k(x_n)\right)\right|
\le K_nr_k^n\sum_T\prod_{\{i,j\}\in T}w_k(x_i-x_j) ω k T ( φ k ( x 1 ) ; … ; φ k ( x n ) ) ≤ K n r k n T ∑ { i , j } ∈ T ∏ w k ( x i − x j ) for all x 1 , … , x n ∈ Z 4 x_1,\ldots,x_n\in\mathbb{Z}^4 x 1 , … , x n ∈ Z 4 T T T { 1 , … , n } \{1,\ldots,n\} { 1 , … , n }
(T2) Saturation. C k ( 0 ) ≥ c r k 2 a k W k C_k(0)\ge cr_k^2a_kW_k C k ( 0 ) ≥ c r k 2 a k W k c > 0 c>0 c > 0 k k k
(T3) Physical volume. B ∗ = sup k a k 4 W k < ∞ B_*=\sup_ka_k^4W_k<\infty B ∗ = sup k a k 4 W k < ∞
The amplitude r k r_k r k Q ⊂ R 4 Q\subset\mathbb{R}^4 Q ⊂ R 4 ∣ Q ∣ k = a k 4 # { x ∈ Z 4 : a k x ∈ Q } |Q|_k=a_k^4\#\{x\in\mathbb{Z}^4:a_kx\in Q\} ∣ Q ∣ k = a k 4 # { x ∈ Z 4 : a k x ∈ Q } L L L ( L + a 0 ) 4 (L+a_0)^4 ( L + a 0 ) 4
Lemma 55. Under (T1), for n ≥ 2 n\ge2 n ≥ 2 f 1 , … , f n f_1,\ldots,f_n f 1 , … , f n f 1 f_1 f 1 Q Q Q
∣ ω k T ( A k ( f 1 ) ; … ; A k ( f n ) ) ∣ ≤ K n n n − 2 r k n ∣ Q ∣ k ( a k 4 W k ) n − 1 ∏ j = 1 n ∥ f j ∥ ∞ . \left|\omega_k^{\mathrm{T}}\left(A_k(f_1);\ldots;A_k(f_n)\right)\right|
\le K_nn^{n-2}r_k^n|Q|_k(a_k^4W_k)^{n-1}\prod_{j=1}^n\|f_j\|_\infty. ω k T ( A k ( f 1 ) ; … ; A k ( f n ) ) ≤ K n n n − 2 r k n ∣ Q ∣ k ( a k 4 W k ) n − 1 j = 1 ∏ n ∥ f j ∥ ∞ . Proof. Joint cumulants of order n ≥ 2 n\ge2 n ≥ 2 a k 4 n ∑ x 1 , … , x n ∏ j ∣ f j ( a k x j ) ∣ a_k^{4n}\sum_{x_1,\ldots,x_n}\prod_j|f_j(a_kx_j)| a k 4 n ∑ x 1 , … , x n ∏ j ∣ f j ( a k x j ) ∣ T T T x 1 x_1 x 1 T T T W k W_k W k x 1 x_1 x 1 ∣ Q ∣ k a k − 4 |Q|_ka_k^{-4} ∣ Q ∣ k a k − 4 n n − 2 n^{n-2} n n − 2
Theorem 56. Assume (T1)–(T3).
(i) For n ≥ 2 n\ge2 n ≥ 2 f j f_j f j
∣ ω k T ( Z k A k ( f 1 ) ; … ; Z k A k ( f n ) ) ∣ ≤ K n n n − 2 c − n / 2 B ∗ n / 2 − 1 ∣ Q ∣ k ∏ j = 1 n ∥ f j ∥ ∞ , \left|\omega_k^{\mathrm{T}}\left(Z_kA_k(f_1);\ldots;Z_kA_k(f_n)\right)\right|
\le K_nn^{n-2}c^{-n/2}B_*^{n/2-1}|Q|_k\prod_{j=1}^n\|f_j\|_\infty, ω k T ( Z k A k ( f 1 ) ; … ; Z k A k ( f n ) ) ≤ K n n n − 2 c − n /2 B ∗ n /2 − 1 ∣ Q ∣ k j = 1 ∏ n ∥ f j ∥ ∞ , and the moments ω k ( ∏ j Z k A k ( f j ) ) \omega_k\left(\prod_jZ_kA_k(f_j)\right) ω k ( ∏ j Z k A k ( f j ) ) f j f_j f j Q Q Q M n ( Q ) ∏ j ∥ f j ∥ ∞ M_n(Q)\prod_j\|f_j\|_\infty M n ( Q ) ∏ j ∥ f j ∥ ∞ M n ( Q ) M_n(Q) M n ( Q ) k k k
(ii) Every sequence of k k k
S n ( f 1 , … , f n ) = lim k ω k ( ∏ j = 1 n Z k A k ( f j ) ) S_n(f_1,\ldots,f_n)=\lim_k\omega_k\left(\prod_{j=1}^nZ_kA_k(f_j)\right) S n ( f 1 , … , f n ) = k lim ω k ( j = 1 ∏ n Z k A k ( f j ) ) exists for all n n n f j ∈ C c ( R 4 ) f_j\in C_c(\mathbb{R}^4) f j ∈ C c ( R 4 ) S n S_n S n M n ( Q ) ∏ j ∥ f j ∥ ∞ M_n(Q)\prod_j\|f_j\|_\infty M n ( Q ) ∏ j ∥ f j ∥ ∞ Q Q Q ( R 4 ) n (\mathbb{R}^4)^n ( R 4 ) n R 4 \mathbb{R}^4 R 4 ω k \omega_k ω k K K K Theorem 48 along the subsequence is
K ( y , y ′ ) = S 2 ( f 0 ( ⋅ − y ) ∘ θ , f 0 ( ⋅ − y ′ ) ) , K(y,y')=S_2\left(f_0(\cdot-y)\circ\theta,f_0(\cdot-y')\right), K ( y , y ′ ) = S 2 ( f 0 ( ⋅ − y ) ∘ θ , f 0 ( ⋅ − y ′ ) ) , with θ \theta θ
(iii) Put S 0 = 1 S_0=1 S 0 = 1 S 1 = 0 S_1=0 S 1 = 0 θ \theta θ t ↦ − t t\mapsto-t t ↦ − t R 4 \mathbb{R}^4 R 4 c α ∈ C c_\alpha\in\mathbb{C} c α ∈ C f α , j ∈ C c ( R 4 ) f_{\alpha,j}\in C_c(\mathbb{R}^4) f α , j ∈ C c ( R 4 ) { t > 0 } \{t>0\} { t > 0 }
∑ α , β c ˉ α c β S ( ∏ j f α , j ‾ ∘ θ ⋅ ∏ j f β , j ) ≥ 0 , \sum_{\alpha,\beta}\bar c_\alpha c_\beta S\left(\prod_j\overline{f_{\alpha,j}}\circ\theta\cdot\prod_jf_{\beta,j}\right)\ge0, α , β ∑ c ˉ α c β S ( j ∏ f α , j ∘ θ ⋅ j ∏ f β , j ) ≥ 0 , where S S S n n n S n S_n S n H \mathcal{H} H Ω \Omega Ω Ψ ( P ) \Psi(P) Ψ ( P ) P P P ⟨ Ψ ( P ) , Ψ ( P ′ ) ⟩ = S ( ϑ P ‾ ⋅ P ′ ) \langle\Psi(P),\Psi(P')\rangle=S(\vartheta\overline{P}\cdot P') ⟨ Ψ ( P ) , Ψ ( P ′ )⟩ = S ( ϑ P ⋅ P ′ ) Ψ ( P 0 ) \Psi(P_0) Ψ ( P 0 ) f 0 f_0 f 0 Ω \Omega Ω
(iv) Suppose the ω k \omega_k ω k t ≥ 0 t\geq0 t ≥ 0 H \mathcal{H} H e − t H e^{-tH} e − t H H ≥ 0 H\geq0 H ≥ 0 H H H σ > 0 \sigma>0 σ > 0 C k ( σ ) ≥ 1 2 C k ( 0 ) C_k(\sigma)\geq\frac{1}{2}C_k(0) C k ( σ ) ≥ 2 1 C k ( 0 ) k k k Theorem 48 is nontrivial.
(v) If a k 4 W k → 0 a_k^4W_k\to0 a k 4 W k → 0 S n S_n S n S 2 S_2 S 2 S 2 S_2 S 2
(vi) If the ω k \omega_k ω k Theorem 45 holds, then 0 0 0 H H H σ ( H ) ⊂ { 0 } ∪ [ m ∗ , ∞ ) \sigma(H)\subset\{0\}\cup[m_*,\infty) σ ( H ) ⊂ { 0 } ∪ [ m ∗ , ∞ ) m ∗ = log ( 1 / α ) / ( 2 ℓ ) m_*=\log(1/\alpha)/(2\ell) m ∗ = log ( 1/ α ) / ( 2 ℓ ) Ω ⊥ ≠ 0 \Omega^\perp\neq0 Ω ⊥ = 0
Proof. (i) By (T2), Z k n = C k ( 0 ) − n / 2 Z_k^n=C_k(0)^{-n/2} Z k n = C k ( 0 ) − n /2 ( c r k 2 a k 4 W k ) − n / 2 (c r_k^2a_k^4W_k)^{-n/2} ( c r k 2 a k 4 W k ) − n /2 r k n r_k^n r k n ( a k 4 W k ) n / 2 − 1 ≤ B ∗ n / 2 − 1 (a_k^4W_k)^{n/2-1}\leq B_*^{n/2-1} ( a k 4 W k ) n /2 − 1 ≤ B ∗ n /2 − 1 n ≥ 2 n\geq2 n ≥ 2 { 1 , … , n } \{1,\ldots,n\} { 1 , … , n } A k ( f ) A_k(f) A k ( f ) ∣ Q ∣ k ≤ ( L + a 0 ) 4 |Q|_k\leq(L+a_0)^4 ∣ Q ∣ k ≤ ( L + a 0 ) 4 L L L M n ( Q ) M_n(Q) M n ( Q )
(ii) For each cube Q L = [ − L , L ] 4 Q_L=[-L,L]^4 Q L = [ − L , L ] 4 L ∈ N L\in\mathbb{N} L ∈ N Q L Q_L Q L F \mathcal{F} F ω k ( ∏ j Z k A k ( f j ) ) \omega_k(\prod_j Z_kA_k(f_j)) ω k ( ∏ j Z k A k ( f j )) f j ∈ F f_j\in\mathcal{F} f j ∈ F
∣ ω k ( ∏ j Z k A k ( f j ) ) − ω k ( ∏ j Z k A k ( g j ) ) ∣ ≤ M n ( Q ) ∑ i ∥ f i − g i ∥ ∞ ∏ j ≠ i max ( ∥ f j ∥ ∞ , ∥ g j ∥ ∞ ) \left|\omega_k\left(\prod_jZ_kA_k(f_j)\right)-\omega_k\left(\prod_jZ_kA_k(g_j)\right)\right|
\leq M_n(Q)\sum_i\|f_i-g_i\|_\infty\prod_{j\ne i}\max(\|f_j\|_\infty,\|g_j\|_\infty) ω k ( j ∏ Z k A k ( f j ) ) − ω k ( j ∏ Z k A k ( g j ) ) ≤ M n ( Q ) i ∑ ∥ f i − g i ∥ ∞ j = i ∏ max ( ∥ f j ∥ ∞ , ∥ g j ∥ ∞ ) for functions supported in Q Q Q f j ∈ C c ( R 4 ) f_j\in C_c(\mathbb{R}^4) f j ∈ C c ( R 4 ) C c ∞ ( R 4 ) C_c^\infty(\mathbb{R}^4) C c ∞ ( R 4 ) y y y k k k y / a k ∈ Z 4 y/a_k\in\mathbb{Z}^4 y / a k ∈ Z 4 ω k \omega_k ω k Proposition 47 (i)) give ω k ( ∏ j Z k A k ( f j ( ⋅ − y ) ) ) = ω k ( ∏ j Z k A k ( f j ) ) \omega_k(\prod_jZ_kA_k(f_j(\cdot-y)))=\omega_k(\prod_jZ_kA_k(f_j)) ω k ( ∏ j Z k A k ( f j ( ⋅ − y ))) = ω k ( ∏ j Z k A k ( f j )) y y y y m → y y_m\to y y m → y f j ( ⋅ − y m ) f_j(\cdot-y_m) f j ( ⋅ − y m ) f j ( ⋅ − y ) f_j(\cdot-y) f j ( ⋅ − y ) y y y K K K A k y = A k ( f 0 ( ⋅ − y ) ) A_k^y=A_k(f_0(\cdot-y)) A k y = A k ( f 0 ( ⋅ − y )) n = 2 n=2 n = 2
(iii) Let ϑ \vartheta ϑ θ k \theta_k θ k a k Z 4 a_k\mathbb{Z}^4 a k Z 4 θ \theta θ a k a_k a k Φ \Phi Φ ϑ \vartheta ϑ ϑ ( A k ( f ˉ k ) ) = A k ( f ˉ k ∘ θ k ) \vartheta(A_k(\bar f_k))=A_k(\bar f_k\circ\theta_k) ϑ ( A k ( f ˉ k )) = A k ( f ˉ k ∘ θ k ) P k = ∑ β c β ∏ j Z k A k ( f β , j ) P_k=\sum_\beta c_\beta\prod_jZ_kA_k(f_{\beta,j}) P k = ∑ β c β ∏ j Z k A k ( f β , j ) R k = ∑ α c ˉ α ∏ j Z k A k ( f ˉ α , j ∘ θ ) R_k=\sum_\alpha\bar c_\alpha\prod_jZ_kA_k(\bar f_{\alpha,j}\circ\theta) R k = ∑ α c ˉ α ∏ j Z k A k ( f ˉ α , j ∘ θ ) ϑ ( P ˉ k ) \vartheta(\bar P_k) ϑ ( P ˉ k ) R k R_k R k θ \theta θ θ k \theta_k θ k ω k ( ϑ ( P ˉ k ) R k ) ≥ 0 \omega_k(\vartheta(\bar P_k)R_k)\geq0 ω k ( ϑ ( P ˉ k ) R k ) ≥ 0
∣ ω k ( ϑ ( P ˉ k ) P k ) − ω k ( R k P k ) ∣ \left|\omega_k(\vartheta(\bar P_k)P_k)-\omega_k(R_kP_k)\right| ω k ( ϑ ( P ˉ k ) P k ) − ω k ( R k P k ) k k k max α , j ∥ f α , j ∘ θ k − f α , j ∘ θ ∥ ∞ \max_{\alpha,j}\|f_{\alpha,j}\circ\theta_k-f_{\alpha,j}\circ\theta\|_\infty max α , j ∥ f α , j ∘ θ k − f α , j ∘ θ ∥ ∞ ω k ( R k P k ) \omega_k(R_kP_k) ω k ( R k P k ) H \mathcal{H} H [62 ] . By the same approximation of θ \theta θ θ k \theta_k θ k
∥ Ψ ( P 0 ) ∥ 2 = lim k Z k 2 C k ( 0 ) = 1 , ⟨ Ω , Ψ ( P 0 ) ⟩ = S 1 ( f 0 ) = 0 \|\Psi(P_0)\|^2=\lim_k Z_k^2C_k(0)=1,\qquad\langle\Omega,\Psi(P_0)\rangle=S_1(f_0)=0 ∥Ψ ( P 0 ) ∥ 2 = lim k Z k 2 C k ( 0 ) = 1 , ⟨ Ω , Ψ ( P 0 )⟩ = S 1 ( f 0 ) = 0
(iv) For a polynomial Ψ ( P ) \Psi(P) Ψ ( P ) τ t P \tau_tP τ t P P P P t ≥ 0 t\geq0 t ≥ 0 θ ∘ τ t = τ − t ∘ θ \theta\circ\tau_t=\tau_{-t}\circ\theta θ ∘ τ t = τ − t ∘ θ ⟨ Ψ ( τ t P ) , Ψ ( P ′ ) ⟩ = ⟨ Ψ ( P ) , Ψ ( τ t P ′ ) ⟩ \langle\Psi(\tau_tP),\Psi(P')\rangle=\langle\Psi(P),\Psi(\tau_tP')\rangle ⟨ Ψ ( τ t P ) , Ψ ( P ′ )⟩ = ⟨ Ψ ( P ) , Ψ ( τ t P ′ )⟩ q P ( t ) = ∥ Ψ ( τ t P ) ∥ 2 = ⟨ Ψ ( P ) , Ψ ( τ 2 P ) ⟩ q_P(t)=\|\Psi(\tau_tP)\|^2=\langle\Psi(P),\Psi(\tau_{2}P)\rangle q P ( t ) = ∥Ψ ( τ t P ) ∥ 2 = ⟨ Ψ ( P ) , Ψ ( τ 2 P )⟩ P P P d d d q P ( t ) q_P(t) q P ( t ) d d d ∣ Q ∣ k |Q|_k ∣ Q ∣ k Q Q Q O ( 1 + t ) O(1+t) O ( 1 + t ) q P ( t ) ≤ C P ( 1 + t ) 4 d q_P(t)\leq C_P(1+t)^{4d} q P ( t ) ≤ C P ( 1 + t ) 4 d q P ( t ) ≤ q P ( 0 ) 1 / 2 q P ( 2 t ) 1 / 2 q_P(t)\leq q_P(0)^{1/2}q_P(2t)^{1/2} q P ( t ) ≤ q P ( 0 ) 1/2 q P ( 2 t ) 1/2 q P ( t ) ≤ q P ( 0 ) 2 − m q P ( 2 m t ) 1 − 2 − m q_P(t)\leq q_P(0)^{2^{-m}}q_P(2^mt)^{1-2^{-m}} q P ( t ) ≤ q P ( 0 ) 2 − m q P ( 2 m t ) 1 − 2 − m m m m m → ∞ m\to\infty m → ∞ q P ( t ) ≤ q P ( 0 ) q_P(t)\leq q_P(0) q P ( t ) ≤ q P ( 0 ) τ t \tau_t τ t T ( t ) T(t) T ( t ) H \mathcal{H} H T ( s ) T ( t ) = T ( s + t ) T(s)T(t)=T(s+t) T ( s ) T ( t ) = T ( s + t ) T ( t ) = T ( t / 2 ) 2 ≥ 0 T(t)=T(t/2)^2\geq0 T ( t ) = T ( t /2 ) 2 ≥ 0 ∥ Ψ ( τ t P ) − Ψ ( P ) ∥ → 0 \|\Psi(\tau_tP)-\Psi(P)\|\to0 ∥Ψ ( τ t P ) − Ψ ( P ) ∥ → 0 t → 0 t\to0 t → 0 T ( t ) = e − t H T(t)=e^{-tH} T ( t ) = e − t H H ≥ 0 H\geq0 H ≥ 0 [62 ] . Spatial translations preserve the form, have the opposite translations as inverses, commute with τ t \tau_t τ t
For the ratio let f 0 , t = f 0 ( ⋅ − ( t , 0 ) ) f_{0,t}=f_0(\cdot-(t,0)) f 0 , t = f 0 ( ⋅ − ( t , 0 )) t > 0 t>0 t > 0 k k k τ t / a k A k = A k ( f 0 , t ) \tau_{t/a_k}A_k=A_k(f_{0,t}) τ t / a k A k = A k ( f 0 , t )
C k ( t ) − C k ( 0 ) = ω k ( A k ( f ˉ 0 ∘ θ ) A k ( f 0 , t − f ˉ 0 ) ) . C_k(t)-C_k(0)=\omega_k\bigl(A_k(\bar f_0\circ\theta)A_k(f_{0,t}-\bar f_0)\bigr). C k ( t ) − C k ( 0 ) = ω k ( A k ( f ˉ 0 ∘ θ ) A k ( f 0 , t − f ˉ 0 ) ) . a covariance of two centred fields. The case n = 2 n=2 n = 2 Z k 2 Z_k^2 Z k 2 L 0 ∥ f 0 , t − f 0 ∥ ∞ L_0\|f_{0,t}-f_0\|_\infty L 0 ∥ f 0 , t − f 0 ∥ ∞ L 0 = ( K 2 / c ) sup k ∣ Q ∣ k ∥ f 0 ∥ ∞ L_0=(K_2/c)\sup_k|Q|_k\|f_0\|_\infty L 0 = ( K 2 / c ) sup k ∣ Q ∣ k ∥ f 0 ∥ ∞ Q Q Q f 0 ∘ θ k f_0\circ\theta_k f 0 ∘ θ k k k k f 0 f_0 f 0 σ > 0 \sigma>0 σ > 0 L 0 ∥ f 0 , σ − f 0 ∥ ∞ ≤ 1 2 L_0\|f_{0,\sigma}-f_0\|_\infty\leq\frac12 L 0 ∥ f 0 , σ − f 0 ∥ ∞ ≤ 2 1 C k ( σ ) / C k ( 0 ) ≥ 1 2 C_k(\sigma)/C_k(0)\geq\frac12 C k ( σ ) / C k ( 0 ) ≥ 2 1 Theorem 48 (iv) applies.
(v) For n ≥ 3 n\geq3 n ≥ 3 n / 2 − 1 n/2-1 n /2 − 1 ( a k 4 W k ) n / 2 − 1 (a_k^4W_k)^{n/2-1} ( a k 4 W k ) n /2 − 1 B ∗ n / 2 − 1 B_*^{n/2-1} B ∗ n /2 − 1 n ≥ 3 n\geq3 n ≥ 3
(vi) As in the proof of Theorem 45 , T k T_k T k log ( 1 / α ) / ( 2 n k ) \log(1/\alpha)/(2n_k) log ( 1/ α ) / ( 2 n k ) Ω k \Omega_k Ω k P P P v k = Ψ P k − ω k ( P k ) Ω k v_k=\Psi_{P_k}-\omega_k(P_k)\Omega_k v k = Ψ P k − ω k ( P k ) Ω k P k P_k P k t t t k k k
ω k ( ϑ ( P ˉ k ) τ t / a k P k ) − ∣ ω k ( P k ) ∣ 2 = ⟨ v k , T k t / a k v k ⟩ ≤ e − m ∗ t ∥ v k ∥ 2 , \omega_k\bigl(\vartheta(\bar P_k)\tau_{t/a_k}P_k\bigr)-|\omega_k(P_k)|^2
=\langle v_k,T_k^{t/a_k}v_k\rangle\leq e^{-m_*t}\|v_k\|^2, ω k ( ϑ ( P ˉ k ) τ t / a k P k ) − ∣ ω k ( P k ) ∣ 2 = ⟨ v k , T k t / a k v k ⟩ ≤ e − m ∗ t ∥ v k ∥ 2 , and both sides are finite combinations of moments, which converge by (ii) and the approximation of θ \theta θ ⟨ v , e − t H v ⟩ ≤ e − m ∗ t ∥ v ∥ 2 \langle v,e^{-tH}v\rangle\leq e^{-m_*t}\|v\|^2 ⟨ v , e − t H v ⟩ ≤ e − m ∗ t ∥ v ∥ 2 v = Ψ ( P ) − ⟨ Ω , Ψ ( P ) ⟩ Ω v=\Psi(P)-\langle\Omega,\Psi(P)\rangle\Omega v = Ψ ( P ) − ⟨ Ω , Ψ ( P )⟩ Ω v v v Ω ⊥ \Omega^\perp Ω ⊥ Ω ⊥ \Omega^\perp Ω ⊥ t ≥ 0 t\geq0 t ≥ 0 Theorem 45 . By (iii), Ψ ( P 0 ) \Psi(P_0) Ψ ( P 0 ) Ω ⊥ \Omega^\perp Ω ⊥
Part (iv) shows that the order-two cases of (T1) and (T2) already contain the lower bound on the ratio that Theorem 48 (iv) needs; they are not harmless combinatorial assumptions, and proving them along the continuum tuning is at least as hard as that lower bound. Part (vi) passes the conditional mass gap of Theorem 45 to every polynomial in the smeared fields. Neither part proves Euclidean invariance, independence of the limit from the subsequence, or the reconstruction of the field operators, for which the growth of M n ( Q ) M_n(Q) M n ( Q ) n n n
Remark 57. The amplitude scale cannot be dropped for a bounded field. Suppose Var φ k ( 0 ) → 0 \operatorname{Var}\varphi_k(0) \to0 Var φ k ( 0 ) → 0 ω k \omega_k ω k
C k ( 0 ) ≤ ω k ( ∣ A k ∣ 2 ) ≤ ( a k 4 ∑ x ∣ f 0 ( a k x ) ∣ ) 2 Var φ k ( 0 ) → 0. C_k(0) \leq\omega_k(|A_k|^2) \leq\left(a_k^4 \sum_x |f_0(a_kx)|\right)^2 \operatorname{Var}\varphi_k(0) \to0. C k ( 0 ) ≤ ω k ( ∣ A k ∣ 2 ) ≤ ( a k 4 x ∑ ∣ f 0 ( a k x ) ∣ ) 2 Var φ k ( 0 ) → 0. With r k = 1 r_k=1 r k = 1 a k 4 W k → 0 a_k^4W_k\to0 a k 4 W k → 0 r k 2 r_k^2 r k 2 Var φ k ( 0 ) \operatorname{Var}\varphi_k(0) Var φ k ( 0 ) r k r_k r k
Remark 58. Let 0 < ε < 1 0<\varepsilon<1 0 < ε < 1 m > 0 m>0 m > 0 π \pi π { − 1 , 0 , 1 } \{-1,0,1\} { − 1 , 0 , 1 } ε / 2 \varepsilon/2 ε /2 1 − ε 1-\varepsilon 1 − ε ε / 2 \varepsilon/2 ε /2 P t = e − m t I + ( 1 − e − m t ) Π P_t=e^{-mt}I+(1-e^{-mt})\Pi P t = e − m t I + ( 1 − e − m t ) Π L 2 ( π ) L^2(\pi) L 2 ( π ) Π f = π ( f ) \Pi f=\pi(f) Π f = π ( f ) P t ≥ 0 P_t\geq0 P t ≥ 0 m m m X ( s ) = s / ε X(s)=s/\sqrt{\varepsilon} X ( s ) = s / ε π ( X ) = π ( X 3 ) = 0 \pi(X)=\pi(X^3)=0 π ( X ) = π ( X 3 ) = 0 π ( X 2 ) = 1 \pi(X^2)=1 π ( X 2 ) = 1 π ( X 4 ) = 1 / ε \pi(X^4)=1/\varepsilon π ( X 4 ) = 1/ ε P t X = e − m t X P_tX=e^{-mt}X P t X = e − m t X e − m t e^{-mt} e − m t ε \varepsilon ε t 1 ≤ t 2 ≤ t 3 ≤ t 4 t_1\leq t_2\leq t_3\leq t_4 t 1 ≤ t 2 ≤ t 3 ≤ t 4 q i = e − m ( t i + 1 − t i ) q_i=e^{-m(t_{i+1}-t_i)} q i = e − m ( t i + 1 − t i ) q 3 π ( X P t 2 − t 1 ( X P t 3 − t 2 X 2 ) ) = q 1 q 3 ( q 2 / ε + 1 − q 2 ) q_3\pi(XP_{t_2-t_1}(XP_{t_3-t_2}X^2))=q_1q_3(q_2/\varepsilon+1-q_2) q 3 π ( X P t 2 − t 1 ( X P t 3 − t 2 X 2 )) = q 1 q 3 ( q 2 / ε + 1 − q 2 ) q 1 q 3 ( q 2 / ε − q 2 − 2 q 2 2 ) q_1q_3(q_2/\varepsilon-q_2-2q_2^2) q 1 q 3 ( q 2 / ε − q 2 − 2 q 2 2 ) ε → 0 \varepsilon\to0 ε → 0 ε \varepsilon ε
The hypotheses (T1)–(T3) are uniform cluster estimates for the continuum tuning β k → ∞ \beta_k\to\infty β k → ∞ a k a_k a k m k ∈ [ 0 , ∞ ] m_k\in[0,\infty] m k ∈ [ 0 , ∞ ] T k T_k T k Ω k \Omega_k Ω k ∥ T k v ∥ ≤ e − m k ∥ v ∥ \lVert T_kv\rVert\leq e^{-m_k}\lVert v\rVert ∥ T k v ∥ ≤ e − m k ∥ v ∥ v ⊥ Ω k v\perp\Omega_k v ⊥ Ω k t 0 > 0 t_0>0 t 0 > 0 f 0 f_0 f 0
Proposition 59. (i) For every dyadic σ > 0 \sigma>0 σ > 0 k k k C k ( σ ) ≤ e − m k σ / a k C k ( 0 ) C_k(\sigma)\leq e^{-m_k\sigma/a_k}C_k(0) C k ( σ ) ≤ e − m k σ / a k C k ( 0 ) m k / a k → ∞ m_k/a_k\to\infty m k / a k → ∞ Theorem 48 is trivial; this is the case for β k = β \beta_k=\beta β k = β Theorem 5 , where m k ≥ m G ( β ) m_k\geq m_G(\beta) m k ≥ m G ( β )
(ii) Suppose the ω k \omega_k ω k n = 2 n=2 n = 2 m k ≤ a k log 2 / σ m_k\leq a_k\log2/\sigma m k ≤ a k log 2/ σ k k k σ \sigma σ K 2 / c K_2/c K 2 / c f 0 f_0 f 0 Theorem 45 holds, then m ∗ a k ≤ m k ≤ a k log 2 / σ m_*a_k\leq m_k\leq a_k\log2/\sigma m ∗ a k ≤ m k ≤ a k log 2/ σ m ∗ ≤ log 2 / σ m_*\leq\log2/\sigma m ∗ ≤ log 2/ σ
(iii) Under (T1) for n = 2 n=2 n = 2 Q Q Q f 0 f_0 f 0 γ = c / ( K 2 ∥ f 0 ∥ ∞ 2 sup k ∣ Q k ∣ ) \gamma=c/(K_2\lVert f_0\rVert_\infty^2\sup_k|Q_k|) γ = c / ( K 2 ∥ f 0 ∥ ∞ 2 sup k ∣ Q k ∣ ) k k k
∑ z ∈ Z 4 ∣ z 0 ∣ ≥ R k w k ( z ) ≥ γ W k , R k = 2 t 0 a k − 1. \sum_{\substack{z\in\mathbb{Z}^4\\ |z_0|\geq R_k}} w_k(z)\geq\gamma W_k,\qquad R_k=\frac{2t_0}{a_k}-1. z ∈ Z 4 ∣ z 0 ∣ ≥ R k ∑ w k ( z ) ≥ γ W k , R k = a k 2 t 0 − 1. If w k ( z ) = e − μ k ∣ z 0 ∣ u k ( z ˉ ) w_k(z)=e^{-\mu_k|z_0|}u_k(\bar z) w k ( z ) = e − μ k ∣ z 0 ∣ u k ( z ˉ ) z ˉ \bar z z ˉ z z z u k ≥ 0 u_k\geq0 u k ≥ 0 Z 3 \mathbb{Z}^3 Z 3 μ k R k ≤ log ( 2 / γ ) \mu_kR_k\leq\log(2/\gamma) μ k R k ≤ log ( 2/ γ ) log ( 2 / γ ) / ( 2 t 0 − a k ) \log(2/\gamma)/(2t_0-a_k) log ( 2/ γ ) / ( 2 t 0 − a k )
Proof. (i) In the proof of Theorem 48 , C k ( σ ) / C k ( 0 ) = ⟨ Ψ k , T k σ / a k Ψ k ⟩ C_k(\sigma)/C_k(0)=\langle\Psi_k,T_k^{\sigma/a_k}\Psi_k\rangle C k ( σ ) / C k ( 0 ) = ⟨ Ψ k , T k σ / a k Ψ k ⟩ Ψ k \Psi_k Ψ k Ω k \Omega_k Ω k T k T_k T k Ω k \Omega_k Ω k m k / a k → ∞ m_k/a_k\to\infty m k / a k → ∞ ρ ( σ ) = 0 \rho(\sigma)=0 ρ ( σ ) = 0 Theorem 48 (iv). At fixed β \beta β Theorem 5 the gap is at least m G ( β ) > 0 m_G(\beta)>0 m G ( β ) > 0 a k → 0 a_k\to0 a k → 0 Proposition 47 (iv)).
(ii) Theorem 56(iv) uses (T1) and (T2) only for n = 2 n=2 n = 2 n / 2 − 1 n/2-1 n /2 − 1 C k ( σ ) ≥ 1 2 C k ( 0 ) C_k(\sigma)\geq\frac12C_k(0) C k ( σ ) ≥ 2 1 C k ( 0 ) k k k e − m k σ / a k ≥ 1 2 e^{-m_k\sigma/a_k}\geq\frac12 e − m k σ / a k ≥ 2 1 Theorem 45 , the proof of that theorem gives m k ≥ log ( 1 / α ) / ( 2 n k ) ≥ m ∗ a k m_k\geq\log(1/\alpha)/(2n_k)\geq m_*a_k m k ≥ log ( 1/ α ) / ( 2 n k ) ≥ m ∗ a k
(iii) C k ( 0 ) = ω k ( A k ( f ˉ 0 ∘ θ k ) A k ( f 0 ) ) C_k(0)=\omega_k(A_k(\bar f_0\circ\theta_k)A_k(f_0)) C k ( 0 ) = ω k ( A k ( f ˉ 0 ∘ θ k ) A k ( f 0 )) x x x a k x 0 ≤ − t 0 + a k a_kx_0\leq-t_0+a_k a k x 0 ≤ − t 0 + a k y y y a k y 0 ≥ t 0 a_ky_0\geq t_0 a k y 0 ≥ t 0 ∣ x 0 − y 0 ∣ ≥ R k |x_0-y_0|\geq R_k ∣ x 0 − y 0 ∣ ≥ R k n = 2 n=2 n = 2
C k ( 0 ) ≤ K 2 r k 2 ∥ f 0 ∥ ∞ 2 a k 8 ∑ x , y ∈ Q k ∣ x 0 − y 0 ∣ ≥ R k w k ( x − y ) ≤ K 2 r k 2 ∥ f 0 ∥ ∞ 2 ∣ Q k ∣ a k 4 ∑ ∣ z 0 ∣ ≥ R k w k ( z ) , \begin{aligned}
C_k(0)\leq K_2r_k^2\lVert f_0\rVert_\infty^2a_k^8
\sum_{\substack{x,y\in Q_k\\ |x_0-y_0|\geq R_k}}w_k(x-y) \\
&\leq K_2r_k^2\lVert f_0\rVert_\infty^2|Q_k|a_k^4
\sum_{|z_0|\geq R_k}w_k(z),
\end{aligned} C k ( 0 ) ≤ K 2 r k 2 ∥ f 0 ∥ ∞ 2 a k 8 x , y ∈ Q k ∣ x 0 − y 0 ∣ ≥ R k ∑ w k ( x − y ) ≤ K 2 r k 2 ∥ f 0 ∥ ∞ 2 ∣ Q k ∣ a k 4 ∣ z 0 ∣ ≥ R k ∑ w k ( z ) , and (T2) bounds the left side below by c r k 2 a k 4 W k cr_k^2a_k^4W_k c r k 2 a k 4 W k R ′ = [ R k ] R'= [R_k] R ′ = [ R k ] R ′ ≤ 0 R'\leq0 R ′ ≤ 0
∑ ∣ j ∣ ≥ R ′ e − μ k ∣ j ∣ ∑ j ∈ Z e − μ k ∣ j ∣ = 2 e − μ k R ′ 1 + e − μ k ≤ 2 e − μ k R k , \frac{\sum_{|j|\geq R'}e^{-\mu_k|j|}}{\sum_{j\in\mathbb{Z}}e^{-\mu_k|j|}}
=\frac{2e^{-\mu_kR'}}{1+e^{-\mu_k}}\leq2e^{-\mu_kR_k}, ∑ j ∈ Z e − μ k ∣ j ∣ ∑ ∣ j ∣ ≥ R ′ e − μ k ∣ j ∣ = 1 + e − μ k 2 e − μ k R ′ ≤ 2 e − μ k R k , while the factor ∑ z u k ( z ˉ ) \sum_z u_k(\bar z) ∑ z u k ( z ˉ ) 2 e − μ k R k ≥ γ 2e^{-\mu_kR_k}\geq\gamma 2 e − μ k R k ≥ γ
So reflection positivity and decay do not supply (T2): at fixed β \beta β Theorem 5 the measures are reflection positive with exponential clustering, and C k ( σ ) / C k ( 0 ) → 0 C_k(\sigma)/C_k(0)\to0 C k ( σ ) / C k ( 0 ) → 0 w k w_k w k r k r_k r k [64 ] , can give the tree bounds along the tuning: (T1) and (T2) hold, if at all, only along couplings where the lattice gap closes at least as fast as a k a_k a k K n K_n K n lim inf k a k 4 W k > 0 \liminf_k a_k^4W_k>0 lim inf k a k 4 W k > 0 W k W_k W k coth ( μ k / 2 ) ≥ 2 / μ k \coth(\mu_k/2)\geq2/\mu_k coth ( μ k /2 ) ≥ 2/ μ k 1 / a k 1/a_k 1/ a k τ \sqrt{\tau} τ
The order-two cases of (T1) and (T2) say more about the reference vector than part (iv) of Theorem 56 records: for a smooth f 0 f_0 f 0 μ k \mu_k μ k [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] Theorem 48 , so that C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ k ( E ) C_k(\sigma)/C_k(0)=\int e^{-\sigma E}\,d\mu_k(E) C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ k ( E )
Proposition 60. Suppose the ω k \omega_k ω k n = 2 n=2 n = 2 f 0 f_0 f 0 m ≥ 1 m\ge1 m ≥ 1 Q Q Q f 0 ( ⋅ − ( s , 0 ) ) f_0(\mathord\cdot-(s,0)) f 0 ( ⋅ − ( s , 0 )) 0 ≤ s ≤ m a 0 0\le s\le ma_0 0 ≤ s ≤ m a 0 B m = ( K 2 / c ) sup k ∣ Q ∣ k ∥ ∂ t m f 0 ∥ ∞ 2 B_m=(K_2/c)\sup_k|Q|_k\|\partial_t^m f_0\|_\infty^2 B m = ( K 2 / c ) sup k ∣ Q ∣ k ∥ ∂ t m f 0 ∥ ∞ 2
(i) ∥ ( 1 − T k ) m Ψ k ∥ 2 ≤ B m a k 2 m \|(1-T_k)^m\Psi_k\|^2\le B_m a_k^{2m} ∥ ( 1 − T k ) m Ψ k ∥ 2 ≤ B m a k 2 m k k k Ψ k = Z k Ψ A k \Psi_k=Z_k\Psi_{A_k} Ψ k = Z k Ψ A k
(ii) ∫ ( ( 1 − e − a k E ) / a k ) 2 m d μ k ( E ) ≤ B m \int((1-e^{-a_kE})/a_k)^{2m}\,d\mu_k(E)\le B_m ∫ (( 1 − e − a k E ) / a k ) 2 m d μ k ( E ) ≤ B m μ k ( [ Λ , ∞ ] ) ≤ B m ( a k + Λ − 1 ) 2 m \mu_k([\Lambda,\infty])\le B_m(a_k+\Lambda^{-1})^{2m} μ k ([ Λ , ∞ ]) ≤ B m ( a k + Λ − 1 ) 2 m Λ > 0 \Lambda>0 Λ > 0 μ \mu μ μ k \mu_k μ k μ ( { ∞ } ) = 0 \mu(\{\infty\})=0 μ ({ ∞ }) = 0 ∫ E 2 m d μ ( E ) ≤ B m \int E^{2m}\,d\mu(E)\le B_m ∫ E 2 m d μ ( E ) ≤ B m Ψ ( P 0 ) \Psi(P_0) Ψ ( P 0 ) H m H^m H m ∥ H m Ψ ( P 0 ) ∥ 2 ≤ B m \|H^m\Psi(P_0)\|^2\le B_m ∥ H m Ψ ( P 0 ) ∥ 2 ≤ B m
(iii) If a k B 1 ≤ 1 2 a_k\sqrt{B_1}\le\frac12 a k B 1 ≤ 2 1 C k ( σ ) ≥ e − B 1 σ C k ( 0 ) C_k(\sigma)\ge e^{-\sqrt{B_1}\sigma}C_k(0) C k ( σ ) ≥ e − B 1 σ C k ( 0 ) σ ∈ a k Z ≥ 0 \sigma\in a_k\mathbb{Z}_{\ge0} σ ∈ a k Z ≥ 0 m k m_k m k Proposition 59 satisfies m k ≤ − log ( 1 − a k B 1 ) ≤ 2 B 1 a k m_k\le-\log(1-a_k\sqrt{B_1})\le2\sqrt{B_1}a_k m k ≤ − log ( 1 − a k B 1 ) ≤ 2 B 1 a k
Proof. (i) Let D k f 0 = ∑ j = 0 m ( − 1 ) j ( m j ) f 0 ( ⋅ − ( j a k , 0 ) ) D_kf_0=\sum_{j=0}^m(-1)^j\binom{m}{j}f_0(\mathord\cdot-(ja_k,0)) D k f 0 = ∑ j = 0 m ( − 1 ) j ( j m ) f 0 ( ⋅ − ( j a k , 0 )) T k j Ψ A k = Ψ τ j A k T_k^j\Psi_{A_k}=\Psi_{\tau_jA_k} T k j Ψ A k = Ψ τ j A k τ j A k = A k ( f 0 ( ⋅ − ( j a k , 0 ) ) ) \tau_jA_k=A_k(f_0(\mathord\cdot-(ja_k,0))) τ j A k = A k ( f 0 ( ⋅ − ( j a k , 0 ))) ( 1 − T k ) m Ψ A k = Ψ A k ( D k f 0 ) (1-T_k)^m\Psi_{A_k}=\Psi_{A_k(D_kf_0)} ( 1 − T k ) m Ψ A k = Ψ A k ( D k f 0 ) ω k ( A k ( D k f 0 ∘ θ k ) A k ( D k f 0 ) ) \omega_k(A_k(D_kf_0\circ\theta_k)A_k(D_kf_0)) ω k ( A k ( D k f 0 ∘ θ k ) A k ( D k f 0 )) D k f 0 D_kf_0 D k f 0 Q Q Q m m m a k a_k a k ∥ D k f 0 ∥ ∞ ≤ a k m ∥ ∂ t m f 0 ∥ ∞ \|D_kf_0\|_\infty\le a_k^m\|\partial_t^m f_0\|_\infty ∥ D k f 0 ∥ ∞ ≤ a k m ∥ ∂ t m f 0 ∥ ∞ n = 2 n=2 n = 2 K 2 r k 2 ∣ Q ∣ k a k 4 W k ∥ D k f 0 ∥ ∞ 2 K_2r_k^2|Q|_ka_k^4W_k\|D_kf_0\|_\infty^2 K 2 r k 2 ∣ Q ∣ k a k 4 W k ∥ D k f 0 ∥ ∞ 2 C k ( 0 ) = Z k − 2 C_k(0)=Z_k^{-2} C k ( 0 ) = Z k − 2 c r k 2 a k 4 W k c r_k^2a_k^4W_k c r k 2 a k 4 W k
(ii) By the spectral theorem the left side of (i) is ∫ ( 1 − e − a k E ) 2 m d μ k ( E ) \int(1-e^{-a_kE})^{2m}\,d\mu_k(E) ∫ ( 1 − e − a k E ) 2 m d μ k ( E ) 1 1 1 E = ∞ E=\infty E = ∞ E ≥ Λ E\ge\Lambda E ≥ Λ ( 1 − e − a k E ) / a k ≥ ( 1 − e − a k Λ ) / a k ≥ Λ / ( 1 + a k Λ ) (1-e^{-a_kE})/a_k\ge(1-e^{-a_k\Lambda})/a_k\ge\Lambda/(1+a_k\Lambda) ( 1 − e − a k E ) / a k ≥ ( 1 − e − a k Λ ) / a k ≥ Λ/ ( 1 + a k Λ ) 1 − e − x ≥ x / ( 1 + x ) 1-e^{-x}\ge x/(1+x) 1 − e − x ≥ x / ( 1 + x ) ( Λ , ∞ ] (\Lambda,\infty] ( Λ , ∞ ] [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] μ ( ( Λ , ∞ ] ) ≤ lim inf k μ k ( ( Λ , ∞ ] ) ≤ B m Λ − 2 m \mu((\Lambda,\infty])\le\liminf_k\mu_k((\Lambda,\infty])\le B_m\Lambda^{-2m} μ (( Λ , ∞ ]) ≤ lim inf k μ k (( Λ , ∞ ]) ≤ B m Λ − 2 m μ ( { ∞ } ) = 0 \mu(\{\infty\})=0 μ ({ ∞ }) = 0 χ \chi χ 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 [ 0 , L ] [0,L] [ 0 , L ] E ≤ ( 1 + a k L ) ( 1 − e − a k E ) / a k E\le(1+a_kL)(1-e^{-a_kE})/a_k E ≤ ( 1 + a k L ) ( 1 − e − a k E ) / a k ∫ χ E 2 m d μ = lim k ∫ χ E 2 m d μ k ≤ B m \int\chi E^{2m}\,d\mu=\lim_k\int\chi E^{2m}\,d\mu_k\le B_m ∫ χ E 2 m d μ = lim k ∫ χ E 2 m d μ k ≤ B m χ \chi χ 1 1 1 Ψ ( P 0 ) \Psi(P_0) Ψ ( P 0 ) ⟨ Ψ ( P 0 ) , e − σ H Ψ ( P 0 ) ⟩ = lim k C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ \langle\Psi(P_0),e^{-\sigma H}\Psi(P_0)\rangle=\lim_kC_k(\sigma)/C_k(0)=\int e^{-\sigma E}\,d\mu ⟨ Ψ ( P 0 ) , e − σ H Ψ ( P 0 )⟩ = lim k C k ( σ ) / C k ( 0 ) = ∫ e − σ E d μ σ > 0 \sigma>0 σ > 0 H H H μ \mu μ
(iii) By (i) with m = 1 m=1 m = 1 ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ a k B 1 \langle\Psi_k,(1-T_k)\Psi_k\rangle\le a_k\sqrt{B_1} ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ a k B 1 C = B 1 C=\sqrt{B_1} C = B 1 Ψ k \Psi_k Ψ k Ω k \Omega_k Ω k 1 − e − m k 1-e^{-m_k} 1 − e − m k
Part (iii) makes Proposition 59 (ii) explicit: the order-two tree and saturation bounds pin the lattice gap below 2 B 1 a k 2\sqrt{B_1}a_k 2 B 1 a k 2 B 1 2\sqrt{B_1} 2 B 1 Theorem 48 (iv) at every separation. The open input of Theorem 48 is therefore implied by these two bounds for one smooth f 0 f_0 f 0
This section states the two unproved inputs of Sec. X, the cube estimate and the limit, and measures the first: what can be controlled uniformly in the coupling, and what any proof of the cube estimate has to produce.
A The two open statements
Theorem 45 reduces a continuum mass gap to two statements, neither of which is proved here or, to our knowledge, anywhere else.
(D) The cube estimate. There are α < 1 \alpha<1 α < 1 ℓ > 0 \ell>0 ℓ > 0 D ( n , α ) D(n,\alpha) D ( n , α ) n a ≤ ℓ na\le\ell na ≤ ℓ β = β ( a ) → ∞ \beta=\beta(a)\to\infty β = β ( a ) → ∞ a a a
(E) The limit. The renormalised Schwinger functions converge along a k = 2 − k a 0 a_k=2^{-k}a_0 a k = 2 − k a 0
(D) and (E) together give the mass gap of the continuum theory; the remaining Clay requirements, Euclidean invariance and the local structure of the limit [47 ] , belong to (E). (D) is a statement about a single cube of fixed physical size and a single coupling at each spacing; (E) is the construction problem, on which Bałaban’s renormalisation group analysis [5 , 6 , 7 ] gives the ultraviolet part in finite volume.
Remark 61. The following count relates (D) to a correlation length. Suppose that the gauge-fixed field responds to a change of one link on the boundary of a cube through a Poisson kernel with decay length ξ \xi ξ n ξ n\xi n ξ K c K_c K c n ξ 3 n\xi^3 n ξ 3 D ( n , α ) D(n,\alpha) D ( n , α ) n ≥ ξ / α n\ge\xi/\alpha n ≥ ξ / α ℓ ≥ ξ / a \ell\ge\xi/a ℓ ≥ ξ / a ξ \xi ξ
The count of Remark 61 can be carried out exactly for a Gaussian field, the quadratic approximation of the gauge-fixed theory at weak coupling, and shows what (D) asks of the gauge-fixed variables.
Proposition 62. Let d ≥ 1 d\ge1 d ≥ 1 b > 0 b>0 b > 0 t ≥ 0 t\ge0 t ≥ 0 n ≥ 2 n\ge2 n ≥ 2 V = [ 0 , n ) d ∩ Z d V=[0,n)^d\cap\mathbb{Z}^d V = [ 0 , n ) d ∩ Z d η ∈ R Z d ∖ V \eta\in\mathbb{R}^{\mathbb{Z}^d\setminus V} η ∈ R Z d ∖ V γ ( ⋅ ∣ η ) \gamma(\mathord\cdot\mid\eta) γ ( ⋅ ∣ η ) R V \mathbb{R}^V R V
exp ( − b 2 [ ∑ { x , y } ∩ V ≠ ∅ ( ϕ x − ϕ y ) 2 + t ∑ x ∈ V ϕ x 2 ] ) , \exp\left(-\frac{b}{2}\left[\sum_{\{x,y\}\cap V\ne\varnothing}(\phi_x-\phi_y)^2+t\sum_{x\in V}\phi_x^2\right]\right), exp − 2 b { x , y } ∩ V = ∅ ∑ ( ϕ x − ϕ y ) 2 + t x ∈ V ∑ ϕ x 2 , the first sum over nearest-neighbour pairs, with ϕ = η \phi=\eta ϕ = η V V V C V , j C_{V,j} C V , j c c c K c K_c K c d V ( ϕ , ϕ ′ ) = ∑ x ∈ V ∣ ϕ x − ϕ x ′ ∣ d_V(\phi,\phi')=\sum_{x\in V}|\phi_x-\phi'_x| d V ( ϕ , ϕ ′ ) = ∑ x ∈ V ∣ ϕ x − ϕ x ′ ∣ δ ( η j , η j ′ ) = ∣ η j − η j ′ ∣ \delta(\eta_j,\eta'_j)=|\eta_j-\eta'_j| δ ( η j , η j ′ ) = ∣ η j − η j ′ ∣ N n = ( n / 2 ) d N_n=(n/2)^d N n = ( n /2 ) d K c = α n ( t ) N n K_c=\alpha_n(t)N_n K c = α n ( t ) N n c c c
α n ( t ) = 1 − t n d 1 T ( t + L n ) − 1 1 \alpha_n(t)=1-\frac{t}{n^d}\mathbf{1}^{\mathsf T}(t+L_n)^{-1}\mathbf{1} α n ( t ) = 1 − n d t 1 T ( t + L n ) − 1 1 and L n L_n L n V V V
for t = 0 t=0 t = 0 K c = N n K_c=N_n K c = N n n n n b b b D ( n , α ) D(n,\alpha) D ( n , α ) α < 1 \alpha<1 α < 1
for t > 0 t>0 t > 0 1 − α n ( t ) ≥ ( 5 / 6 ) d t n 2 / ( t n 2 + 10 d ) 1-\alpha_n(t)\ge(5/6)^d\,tn^2/(tn^2+10d) 1 − α n ( t ) ≥ ( 5/6 ) d t n 2 / ( t n 2 + 10 d ) d = 4 d=4 d = 4 t = m 2 a 2 t=m^2a^2 t = m 2 a 2 ℓ / 2 ≤ n a ≤ ℓ \ell/2\le na\le\ell ℓ /2 ≤ na ≤ ℓ D ( n , α ) D(n,\alpha) D ( n , α ) α = 1 − ( 5 / 6 ) 4 ( m ℓ ) 2 / ( ( m ℓ ) 2 + 160 ) \alpha=1-(5/6)^4(m\ell)^2/((m\ell)^2+160) α = 1 − ( 5/6 ) 4 ( m ℓ ) 2 / (( m ℓ ) 2 + 160 ) a a a
Proof. Let A = t + L n A=t+L_n A = t + L n G = A − 1 G=A^{-1} G = A − 1 B B B B x j = 1 B_{xj}=1 B x j = 1 x ∈ V x\in V x ∈ V j ∉ V j\notin V j ∈ / V γ V ( ⋅ ∣ η ) \gamma_V(\cdot\mid\eta) γ V ( ⋅ ∣ η ) b − 1 G b^{-1}G b − 1 G G B η GB\eta GB η A A A G ≥ 0 G\ge0 G ≥ 0 η j \eta_j η j Δ \Delta Δ Δ G B j \Delta GB_j Δ G B j B j B_j B j j j j W 1 d V ≤ ∣ Δ ∣ ∑ x ( G B j ) x W_1^{d_V}\le|\Delta|\sum_x(GB_j)_x W 1 d V ≤ ∣Δ∣ ∑ x ( G B j ) x ∑ x ϕ x \sum_x\phi_x ∑ x ϕ x C V , j = 1 T G B j C_{V,j}=\mathbf{1}^{\mathsf T}GB_j C V , j = 1 T G B j B 1 = L n 1 B\mathbf{1}=L_n\mathbf{1} B 1 = L n 1
∑ j ∉ V C V , j = 1 T G L n 1 = 1 T 1 − t 1 T G 1 = n d α n ( t ) . \sum_{j\notin V}C_{V,j}=\mathbf{1}^{\mathsf T}GL_n\mathbf{1}
=\mathbf{1}^{\mathsf T}\mathbf{1}-t\mathbf{1}^{\mathsf T}G\mathbf{1}
=n^d\alpha_n(t). j ∈ / V ∑ C V , j = 1 T G L n 1 = 1 T 1 − t 1 T G 1 = n d α n ( t ) . The reflection x i ↦ n − 1 − x i x_i\mapsto n-1-x_i x i ↦ n − 1 − x i V V V n n n x i x_i x i 2 d 2^d 2 d n d α n ( t ) / 2 d = α n ( t ) N n n^d\alpha_n(t)/2^d=\alpha_n(t)N_n n d α n ( t ) / 2 d = α n ( t ) N n
For (ii) let v ( i ) = ( i + 1 ) ( n − i ) v(i)=(i+1)(n-i) v ( i ) = ( i + 1 ) ( n − i ) 0 ≤ i < n 0\le i<n 0 ≤ i < n ψ ( x ) = ∏ i v ( x i ) \psi(x)=\prod_i v(x_i) ψ ( x ) = ∏ i v ( x i ) v v v 2 2 2 s 1 = ∑ i v ( i ) = n ( n + 1 ) ( n + 2 ) / 6 s_1=\sum_i v(i)=n(n+1)(n+2)/6 s 1 = ∑ i v ( i ) = n ( n + 1 ) ( n + 2 ) /6 s 2 = ∑ i v ( i ) 2 = n ( n + 1 ) ( n + 2 ) ( n 2 + 2 n + 2 ) / 30 s_2=\sum_i v(i)^2=n(n+1)(n+2)(n^2+2n+2)/30 s 2 = ∑ i v ( i ) 2 = n ( n + 1 ) ( n + 2 ) ( n 2 + 2 n + 2 ) /30
1 T ψ = s 1 d , ψ T A ψ = t s 2 d + 2 d s 1 s 2 d − 1 . \mathbf{1}^{\mathsf T}\psi=s_1^d,\qquad
\psi^{\mathsf T}A\psi=ts_2^d+2ds_1s_2^{d-1}. 1 T ψ = s 1 d , ψ T A ψ = t s 2 d + 2 d s 1 s 2 d − 1 . By the Cauchy–Schwarz inequality for the form of A − 1 A^{-1} A − 1 1 T G 1 ≥ ( 1 T ψ ) 2 / ψ T A ψ \mathbf{1}^{\mathsf T}G\mathbf{1}\ge(\mathbf{1}^{\mathsf T}\psi)^2/\psi^{\mathsf T}A\psi 1 T G 1 ≥ ( 1 T ψ ) 2 / ψ T A ψ
1 − α n ( t ) ≥ ( s 1 2 n s 2 ) d t t + 2 d s 1 / s 2 ≥ ( 5 6 ) d t n 2 t n 2 + 10 d , 1-\alpha_n(t)\ge\left(\frac{s_1^2}{ns_2}\right)^d\frac{t}{t+2ds_1/s_2}
\ge\left(\frac{5}{6}\right)^d\frac{tn^2}{tn^2+10d}, 1 − α n ( t ) ≥ ( n s 2 s 1 2 ) d t + 2 d s 1 / s 2 t ≥ ( 6 5 ) d t n 2 + 10 d t n 2 , since s 1 2 / ( n s 2 ) = 5 6 ( n 2 + 3 n + 2 ) / ( n 2 + 2 n + 2 ) ≥ 5 / 6 s_1^2/(ns_2)=\frac{5}{6}(n^2+3n+2)/(n^2+2n+2)\ge5/6 s 1 2 / ( n s 2 ) = 6 5 ( n 2 + 3 n + 2 ) / ( n 2 + 2 n + 2 ) ≥ 5/6 2 d s 1 / s 2 = 10 d / ( n 2 + 2 n + 2 ) ≤ 10 d / n 2 2ds_1/s_2=10d/(n^2+2n+2)\le10d/n^2 2 d s 1 / s 2 = 10 d / ( n 2 + 2 n + 2 ) ≤ 10 d / n 2 d = 4 d=4 d = 4 t n 2 = m 2 ( n a ) 2 ≥ ( m ℓ ) 2 / 4 tn^2=m^2(na)^2\ge(m\ell)^2/4 t n 2 = m 2 ( na ) 2 ≥ ( m ℓ ) 2 /4 ( m ℓ ) 2 / ( ( m ℓ ) 2 + 160 ) (m\ell)^2/((m\ell)^2+160) ( m ℓ ) 2 / (( m ℓ ) 2 + 160 )
The Gaussian field is not Yang–Mills theory, and the proposition is a benchmark for the condition, not a statement about the Wilson measure. It shows that the averaged cube condition is met at a fixed physical scale by a field with a mass and fails exactly, at every size and every stiffness, for a massless one. Small local fluctuations alone therefore give nothing: in the massless Gaussian field on Z d \mathbb{Z}^d Z d d ≥ 3 d\ge3 d ≥ 3 ϕ x − ϕ x + e 1 \phi_x-\phi_{x+e_1} ϕ x − ϕ x + e 1 1 / ( d b ) 1/(db) 1/ ( d b ) b → ∞ b\to\infty b → ∞ K c = N n K_c=N_n K c = N n t t t
The quadratic approximation of the forest gauge itself, with the plaquette action in place of the Laplacian, is no better than the massless field of Proposition 62(i).
Proposition 63. Let b > 0 b>0 b > 0 n n n V V V Definition 42 . For real functions φ \varphi φ Z 4 \mathbb{Z}^4 Z 4 F F F ( d φ ) p (d\varphi)_p ( d φ ) p φ \varphi φ p p p η ∈ R A ∖ V \eta\in\mathbb{R}^{A\setminus V} η ∈ R A ∖ V γ ( ⋅ ∣ η ) \gamma(\cdot\mid\eta) γ ( ⋅ ∣ η ) R V \mathbb{R}^V R V
exp ( − b 2 ∑ p ∩ V ≠ ∅ ( d φ ) p 2 ) , \exp\left(-\frac{b}{2}\sum_{p\cap V\ne\varnothing}(d\varphi)_p^2\right), exp − 2 b p ∩ V = ∅ ∑ ( d φ ) p 2 , with φ = η \varphi=\eta φ = η A ∖ V A\setminus V A ∖ V φ = 0 \varphi=0 φ = 0 F F F C V , j C_{V,j} C V , j K c K_c K c d V ( φ , φ ′ ) = ∑ i ∈ V ∣ φ i − φ i ′ ∣ d_V(\varphi,\varphi')=\sum_{i\in V}|\varphi_i-\varphi'_i| d V ( φ , φ ′ ) = ∑ i ∈ V ∣ φ i − φ i ′ ∣ δ ( η j , η j ′ ) = ∣ η j − η j ′ ∣ \delta(\eta_j,\eta'_j)=|\eta_j-\eta'_j| δ ( η j , η j ′ ) = ∣ η j − η j ′ ∣
∑ c K c ≥ ∣ V ∣ = 48 N n . \sum_c K_c\ge|V|=48N_n. c ∑ K c ≥ ∣ V ∣ = 48 N n . so K c ≥ N n K_c\ge N_n K c ≥ N n c c c D ( n , α ) D(n,\alpha) D ( n , α ) α < 1 \alpha<1 α < 1 n n n b b b
Proof. The laws. The quadratic form ∑ p ∩ V ≠ ∅ ( d φ ) p 2 \sum_{p\cap V\ne\varnothing}(d\varphi)_p^2 ∑ p ∩ V = ∅ ( d φ ) p 2 φ = 0 \varphi=0 φ = 0 V V V φ = 0 \varphi=0 φ = 0 h h h φ ( x , μ ) = h ( x ) − h ( x + e μ ) \varphi(x,\mu)=h(x)-h(x+e_\mu) φ ( x , μ ) = h ( x ) − h ( x + e μ ) h h h F F F γ ( ⋅ ∣ η ) \gamma(\cdot\mid\eta) γ ( ⋅ ∣ η ) η \eta η ∑ j η j u j \sum_j\eta_j u_j ∑ j η j u j u j ∈ R V u_j\in\mathbb{R}^V u j ∈ R V t = ∑ j ( η j ′ − η j ) u j t=\sum_j(\eta'_j-\eta_j)u_j t = ∑ j ( η j ′ − η j ) u j ∑ i sign ( t i ) φ i \sum_i\operatorname{sign}(t_i)\varphi_i ∑ i sign ( t i ) φ i ∥ t ∥ 1 \|t\|_1 ∥ t ∥ 1 C V , j = ∥ u j ∥ 1 C_{V,j}=\|u_j\|_1 C V , j = ∥ u j ∥ 1
W 1 d V ( γ ( ⋅ ∣ η ) , γ ( ⋅ ∣ η ′ ) ) ≤ ∑ j C V , j ∣ η j ′ − η j ∣ . W_1^{d_V}\bigl(\gamma(\cdot\mid\eta),\gamma(\cdot\mid\eta')\bigr)
\le\sum_j C_{V,j}|\eta'_j-\eta_j|. W 1 d V ( γ ( ⋅ ∣ η ) , γ ( ⋅ ∣ η ′ ) ) ≤ j ∑ C V , j ∣ η j ′ − η j ∣. Lines. By Lemma 9 (i) the lines of F F F ( x , μ ) (x,\mu) ( x , μ ) x x x σ ( x ) ≠ μ \sigma(x)\ne\mu σ ( x ) = μ ℓ \ell ℓ h ℓ h_\ell h ℓ ( d h ℓ ) ( x , μ ) = h ℓ ( x ) − h ℓ ( x + e μ ) (dh_\ell)_{(x,\mu)}=h_\ell(x)-h_\ell(x+e_\mu) ( d h ℓ ) ( x , μ ) = h ℓ ( x ) − h ℓ ( x + e μ ) d h ℓ dh_\ell d h ℓ F F F φ ↦ φ + d h ℓ \varphi\mapsto\varphi+dh_\ell φ ↦ φ + d h ℓ γ ( ⋅ ∣ d h ℓ ) \gamma(\cdot\mid dh_\ell) γ ( ⋅ ∣ d h ℓ ) γ ( ⋅ ∣ 0 ) \gamma(\cdot\mid0) γ ( ⋅ ∣ 0 ) d h ℓ dh_\ell d h ℓ V V V
∑ i ∈ V ∣ ( d h ℓ ) i ∣ ≤ ∑ j ∈ A ∖ V C V , j ∣ ( d h ℓ ) j ∣ . \sum_{i\in V}|(dh_\ell)_i|\le\sum_{j\in A\setminus V}C_{V,j}|(dh_\ell)_j|. i ∈ V ∑ ∣ ( d h ℓ ) i ∣ ≤ j ∈ A ∖ V ∑ C V , j ∣ ( d h ℓ ) j ∣. Here ∣ ( d h ℓ ) e ∣ = 1 |(dh_\ell)_e|=1 ∣ ( d h ℓ ) e ∣ = 1 e e e ℓ \ell ℓ 0 0 0 ∑ ℓ ∣ ( d h ℓ ) e ∣ = 2 \sum_\ell|(dh_\ell)_e|=2 ∑ ℓ ∣ ( d h ℓ ) e ∣ = 2 e e e 2 ∣ V ∣ ≤ 2 ∑ j C V , j = ∑ c K c 2|V|\le2\sum_jC_{V,j}=\sum_cK_c 2∣ V ∣ ≤ 2 ∑ j C V , j = ∑ c K c □ \square □
The proof uses only that gauge transformations constant on the lines of F F F n ≤ 8 n \le8 n ≤ 8 N n N_n N n n n n N n N_n N n
Remark 64. Five other routes stop at (D) or (E), or earlier.
(a) Disjoint blocks. The block criterion of Lemma 114 with a partition into cubes of side n n n ξ \xi ξ 1 / a 1/a 1/ a N n N_n N n 1 / n 1/n 1/ n
(b) Rescaling a fixed lattice gap. Any bound uniform in lattice units, such as those of Theorems 1 –7 , gives trivial continuum two-point functions [15 ] ; rescaling it produces no gap.
(c) Assuming the gap. Hypothesis R of [15 ] assumes the gap in physical units together with the limit. Theorem 45 replaces the gap part by the cube estimate (D), which concerns one box and one coupling at each spacing.
(d) Computation. At a fixed coupling and a fixed small n n n D ( n , α ) D(n,\alpha) D ( n , α ) n = 2 n=2 n = 2 S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) n → ∞ n \to\infty n → ∞
(e) Axioms alone. Proposition 41 shows that positivity, invariance and regularity do not force a gap, and Proposition 40 shows that the growth route to rotation invariance is closed for the action density. The infrared input therefore has to be an estimate such as (D).
Some local quantities of the Wilson measure can be controlled uniformly in β W \beta_W β W β W 2 \beta_W^2 β W 2
Let G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) b = β W b=\beta_W b = β W e b S e^{bS} e b S S = ∑ p N − 1 Re Tr Q p S=\sum_p N^{-1}\operatorname{Re}\operatorname{Tr}Q_p S = ∑ p N − 1 Re Tr Q p g \mathfrak{g} g ⟨ T , R ⟩ = − Tr ( T R ) \langle T,R\rangle=-\operatorname{Tr}(TR) ⟨ T , R ⟩ = − Tr ( T R ) e e e S e S_e S e e e e X T X_T X T U e ↦ e t U U e U_e\mapsto e^{tU}U_e U e ↦ e t U U e u = 0 u=0 u = 0 ∇ e \nabla_e ∇ e Δ e \Delta_e Δ e U e U_e U e q = N 2 − 1 q=N^2-1 q = N 2 − 1 c N = ( N 2 − 1 ) / N c_N=(N^2-1)/N c N = ( N 2 − 1 ) / N
E e = b ∣ ∇ e S e ∣ 2 , W e = E e − c N S e = b ∣ ∇ e S e ∣ 2 + Δ e S e . E_e=b\lvert\nabla_eS_e\rvert^2,\qquad W_e=E_e-c_NS_e=b\lvert\nabla_eS_e\rvert^2+\Delta_eS_e. E e = b ∣ ∇ e S e ∣ 2 , W e = E e − c N S e = b ∣ ∇ e S e ∣ 2 + Δ e S e . Below, ω \omega ω b > 0 b>0 b > 0 e e e U e U_e U e e b S e d U e e^{bS_e}\,dU_e e b S e d U e d U e dU_e d U e
Proposition 65. (i) *For every unit T ∈ g T\in\mathfrak{g} T ∈ g t ∈ R t\in\mathbb{R} t ∈ R 0 ≤ η < N / 12 0\le\eta<N/12 0 ≤ η < N /12
E ω [ e t b X T S e ∣ U e c ] ≤ e 3 t 2 / N , \mathbb{E}_{\omega}\left[e^{t\sqrt{b}X_TS_e}\mid U_{e^c}\right]\le e^{3t^2/N}, E ω [ e t b X T S e ∣ U e c ] ≤ e 3 t 2 / N , E ω [ e η E e ∣ U e c ] ≤ ( 1 − 12 η / N ) − q / 2 , \mathbb{E}_{\omega}\left[e^{\eta E_e}\mid U_{e^c}\right]\le(1-12\eta/N)^{-q/2}, E ω [ e η E e ∣ U e c ] ≤ ( 1 − 12 η / N ) − q /2 , so E ω E e m ≤ m ! ( 24 / N ) m q m / 2 \mathbb{E}_{\omega}E_e^m\le m!(24/N)^m q^{m/2} E ω E e m ≤ m ! ( 24/ N ) m q m /2 m ≥ 1 m\ge1 m ≥ 1 b b b E ω E e = c N E ω S e \mathbb{E}_{\omega}E_e=c_N\mathbb{E}_{\omega}S_e E ω E e = c N E ω S e
(ii) For a bounded function F F F U e U_e U e E ω [ W e F ] = − E ω [ ∇ e S e , ∇ e F ] \mathbb{E}_{\omega}[W_eF]=-\mathbb{E}_{\omega}[\nabla_eS_e,\nabla_eF] E ω [ W e F ] = − E ω [ ∇ e S e , ∇ e F ] E ω [ W e F ] = 0 \mathbb{E}_{\omega}[W_eF]=0 E ω [ W e F ] = 0 F F F U e U_e U e E ω [ W e W f ] = 0 \mathbb{E}_{\omega}[W_eW_f]=0 E ω [ W e W f ] = 0 e e e f f f
(iii) For a reflection positive as in Appendix § 2.4.6 , and let A = ∑ e h e W e A=\sum_e h_eW_e A = ∑ e h e W e ω ( ϑ ( A ) A ) = 0 \omega(\vartheta(A)A)=0 ω ( ϑ ( A ) A ) = 0 Ψ A = 0 \Psi_A=0 Ψ A = 0 ∑ e h e E e \sum_eh_eE_e ∑ e h e E e c N ∑ e h e S e c_N\sum_eh_eS_e c N ∑ e h e S e
Proof. (i) Along u ↦ e u T U e u\mapsto e^{uT}U_e u ↦ e u T U e e e e N − 1 Re Tr ( e u T V ) N^{-1}\operatorname{Re}\operatorname{Tr}(e^{uT}V) N − 1 Re Tr ( e u T V ) N − 1 Re Tr ( V e − u T ) N^{-1}\operatorname{Re}\operatorname{Tr}(Ve^{-uT}) N − 1 Re Tr ( V e − u T ) V V V N − 1 Re Tr ( T 2 e ± u T V ) N^{-1}\operatorname{Re}\operatorname{Tr}(T^2e^{\pm uT}V) N − 1 Re Tr ( T 2 e ± u T V ) − T 2 ≥ 0 -T^2\ge0 − T 2 ≥ 0 Tr ( − T 2 ) = 1 \operatorname{Tr}(-T^2)=1 Tr ( − T 2 ) = 1 ∣ Tr ( T 2 M ) ∣ ≤ 1 \lvert\operatorname{Tr}(T^2M)\rvert\le1 ∣ Tr ( T 2 M )∣ ≤ 1 M M M S e S_e S e 6 / N 6/N 6/ N u = t / b u=t/\sqrt{b} u = t / b b ( S e ( e t / b U e ) − S e ( U e ) ) ≥ t b X T S e ( U e ) − 3 t 2 / N b(S_e(e^{t/\sqrt{b}U_e})-S_e(U_e))\ge t\sqrt{b}X_TS_e(U_e)-3t^2/N b ( S e ( e t / b U e ) − S e ( U e )) ≥ t b X T S e ( U e ) − 3 t 2 / N e b S e d U e / Z e^{bS_e}\,dU_e/Z e b S e d U e / Z E ω [ e b ( S e ( e t / b U e ) − S e ( U e ) ) ∣ U e c ] = 1 \mathbb{E}_{\omega}[e^{b(S_e(e^{t/\sqrt{b}U_e})-S_e(U_e))}\mid U_{e^c}]=1 E ω [ e b ( S e ( e t / b U e ) − S e ( U e )) ∣ U e c ] = 1 G 0 G_0 G 0 g \mathfrak{g} g ω \omega ω e η ∣ Y ∣ 2 = E G e 2 η ⟨ G 0 , Y ⟩ e^{\eta\lvert Y\rvert^2}=\mathbb{E}_Ge^{\sqrt{2\eta}\langle G_0,Y\rangle} e η ∣ Y ∣ 2 = E G e 2 η ⟨ G 0 , Y ⟩ Y = b ∇ e S e Y=\sqrt{b}\nabla_eS_e Y = b ∇ e S e G 0 G_0 G 0 t = 2 η ∣ G 0 ∣ t=\sqrt{2\eta}\lvert G_0\rvert t = 2 η ∣ G 0 ∣ E e η ∣ G 0 ∣ 2 / N = ( 1 − 12 η / N ) − q / 2 \mathbb{E}e^{\eta\lvert G_0\rvert^2/N}=(1-12\eta/N)^{-q/2} E e η ∣ G 0 ∣ 2 / N = ( 1 − 12 η / N ) − q /2 η = N / 24 \eta=N/24 η = N /24 x m ≤ m ! e x x^m\le m!e^{x} x m ≤ m ! e x ∑ A T A 2 = − c N I \sum_A T_A^2=-c_NI ∑ A T A 2 = − c N I Δ e S e = − c N S e \Delta_eS_e=-c_NS_e Δ e S e = − c N S e e b S e ∇ e S e e^{bS_e}\nabla_eS_e e b S e ∇ e S e G G G
(ii) The integral over U e U_e U e e b S e ∇ e F e^{bS_e}\nabla_eF e b S e ∇ e F e b S e ( W e F + ⟨ ∇ e S e , ∇ e F ⟩ ) e^{bS_e}(W_eF+\langle\nabla_eS_e,\nabla_eF\rangle) e b S e ( W e F + ⟨ ∇ e S e , ∇ e F ⟩) W f W_f W f f f f
(iii) Each W e W_e W e ϑ ( A ) \vartheta(A) ϑ ( A ) e e e ω ( ϑ ( A ) W e ) = 0 \omega(\vartheta(A)W_e)=0 ω ( ϑ ( A ) W e ) = 0 e e e
Proposition 66. Let G = S U ( 2 ) G = \mathrm{SU}(2) G = SU ( 2 ) b ≥ 0 b \ge0 b ≥ 0 ω \omega ω b b b p p p
Var ω ( 1 2 Tr Q p ) ≥ 1 2 19 ( 1 + 6 b ) 2 , E ω [ 1 − 1 2 Tr Q p ] ≥ 1 4 b + 1 . \operatorname{Var}_{\omega}\left(\frac{1}{2}\operatorname{Tr}Q_p\right) \ge\frac{1}{2^{19}(1+6b)^2}, \qquad\mathbb{E}_{\omega}\left[1-\frac{1}{2}\operatorname{Tr}Q_p\right] \ge\frac{1}{4b+1}. Var ω ( 2 1 Tr Q p ) ≥ 2 19 ( 1 + 6 b ) 2 1 , E ω [ 1 − 2 1 Tr Q p ] ≥ 4 b + 1 1 . Proof. Let e e e S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) U e = x U_e=x U e = x e r x ⋅ u e^{rx\cdot u} e r x ⋅ u ∣ u ∣ = 1 |u|=1 ∣ u ∣ = 1 0 ≤ r ≤ 6 b 0\le r\le6b 0 ≤ r ≤ 6 b 1 2 Tr Q p = x ⋅ v \frac{1}{2}\operatorname{Tr}Q_p=x\cdot v 2 1 Tr Q p = x ⋅ v v v v X = x ⋅ u X=x\cdot u X = x ⋅ u T = 1 − X T=1-X T = 1 − X T T T e − r t t ( 2 − t ) e^{-rt}\sqrt{t(2-t)} e − r t t ( 2 − t ) [ 0 , 2 ] [0,2] [ 0 , 2 ] n = 1 + r n=1+r n = 1 + r e − r t ≤ e 2 e − n t e^{-rt}\le e^{2}e^{-nt} e − r t ≤ e 2 e − n t e 2 ∫ 0 2 2 t e − n t d t = e 2 π / 2 n − 3 / 2 < 14 n − 3 / 2 e^2\int_0^2\sqrt{2t}e^{-nt}\,dt=e^2\sqrt{\pi/2}n^{-3/2}<14n^{-3/2} e 2 ∫ 0 2 2 t e − n t d t = e 2 π /2 n − 3/2 < 14 n − 3/2 I = [ 1 4 n , 1 2 n ] I=[\frac{1}{4n},\frac{1}{2n}] I = [ 4 n 1 , 2 n 1 ] 1 2 n − 1 / 2 e − 1 / 2 \frac{1}{2}n^{-1/2}e^{-1/2} 2 1 n − 1/2 e − 1/2 J = [ 1 n , 2 n ] J=[\frac{1}{n},\frac{2}{n}] J = [ n 1 , n 2 ] 1 2 n − 1 / 2 e − 3 / 2 \frac{1}{2}n^{-1/2}e^{-3/2} 2 1 n − 1/2 e − 3/2 P ( T ∈ I ) ≥ 1 224 P(T\in I)\ge\frac{1}{224} P ( T ∈ I ) ≥ 224 1 P ( T ∈ J ) ≥ 1 336 P(T\in J)\ge\frac{1}{336} P ( T ∈ J ) ≥ 336 1 T ′ T' T ′ Var T = 1 2 E ( T − T ′ ) 2 ≥ P ( T ∈ I ) P ( T ∈ J ) ( 2 n ) − 2 > 2 − 19 n − 2 \operatorname{Var}T=\frac{1}{2}\mathbb{E}(T-T')^2\ge P(T\in I)P(T\in J)(2n)^{-2}>2^{-19}n^{-2} Var T = 2 1 E ( T − T ′ ) 2 ≥ P ( T ∈ I ) P ( T ∈ J ) ( 2 n ) − 2 > 2 − 19 n − 2 x x x u u u u ⊥ u^\perp u ⊥ Var X \operatorname{Var}X Var X E ( 1 − X 2 ) / 3 \mathbb{E}(1-X^2)/3 E ( 1 − X 2 ) /3 I I I 1 − X 2 = T ( 2 − T ) ≥ 3 8 n 1-X^2=T(2-T)\ge\frac{3}{8n} 1 − X 2 = T ( 2 − T ) ≥ 8 n 3 E ( 1 − X 2 ) / 3 ≥ 1 1792 n ≥ 2 − 19 n − 2 \mathbb{E}(1-X^2)/3\ge\frac{1}{1792n}\ge2^{-19}n^{-2} E ( 1 − X 2 ) /3 ≥ 1792 n 1 ≥ 2 − 19 n − 2 x ⋅ v x\cdot v x ⋅ v 2 − 19 ( 1 + 6 b ) − 2 2^{-19}(1+6b)^{-2} 2 − 19 ( 1 + 6 b ) − 2 d d X e r X ( 1 − X 2 ) \frac{d}{dX}e^{rX}(1-X^2) d X d e r X ( 1 − X 2 ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] 3 E X = r E ( 1 − X 2 ) ≤ r ( 1 − ( E X ) 2 ) 3\mathbb{E}X=r\mathbb{E}(1-X^2)\le r(1-(\mathbb{E}X)^2) 3 E X = r E ( 1 − X 2 ) ≤ r ( 1 − ( E X ) 2 ) E X ≤ ( 9 + 4 r 2 − 3 ) / ( 2 r ) ≤ 2 r / ( 2 r + 3 ) \mathbb{E}X\le(\sqrt{9+4r^2}-3)/(2r)\le2r/(2r+3) E X ≤ ( 9 + 4 r 2 − 3 ) / ( 2 r ) ≤ 2 r / ( 2 r + 3 ) r > 0 r>0 r > 0 E [ 1 − x ⋅ v ] ≥ 1 − E X ≥ 3 / ( 2 r + 3 ) ≥ 1 / ( 4 b + 1 ) \mathbb{E}[1-x\cdot v]\ge1-\mathbb{E}X\ge3/(2r+3)\ge1/(4b+1) E [ 1 − x ⋅ v ] ≥ 1 − E X ≥ 3/ ( 2 r + 3 ) ≥ 1/ ( 4 b + 1 )
For the blocking statement let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) B ≥ 4 B\ge4 B ≥ 4 ( x , μ ) (x,\mu) ( x , μ ) V ( x , μ ) = ∏ r = 0 B − 1 U ( B x + r μ ^ , μ ) V_{(x,\mu)}=\prod_{r=0}^{B-1}U(Bx+r\hat\mu,\mu) V ( x , μ ) = ∏ r = 0 B − 1 U ( B x + r μ ^ , μ )
Proposition 67. For every Wilson Gibbs state ω \omega ω S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) b > 0 b>0 b > 0 E \mathcal{E} E K K K
∥ Law ω ( ( V e ) e ∈ E ) − m ⊗ K ∥ T V ≤ K 2 e 3 b ( 4 b 4 b + 1 ) B − 3 ≤ K 2 exp ( 3 b − B − 3 4 b + 1 ) . \left\|\operatorname{Law}_{\omega}\left((V_e)_{e\in\mathcal{E}}\right)-m^{\otimes K}\right\|_{\mathrm{TV}}
\le\frac{K}{2}e^{3b}\left(\frac{4b}{4b+1}\right)^{B-3}
\le\frac{K}{2}\exp\left(3b-\frac{B-3}{4b+1}\right). Law ω ( ( V e ) e ∈ E ) − m ⊗ K TV ≤ 2 K e 3 b ( 4 b + 1 4 b ) B − 3 ≤ 2 K exp ( 3 b − 4 b + 1 B − 3 ) . where m m m F F F V e V_e V e ∣ F ∣ ≤ 1 |F|\le1 ∣ F ∣ ≤ 1 ϵ \epsilon ϵ 0 ≤ ω ( ϑ ( F ‾ ) F c ) ≤ 6 ϵ 0\le\omega(\vartheta(\overline{F})F^c)\le6\epsilon 0 ≤ ω ( ϑ ( F ) F c ) ≤ 6 ϵ F c = F − ω ( F ) F^c=F-\omega(F) F c = F − ω ( F )
Proof. Select on each coarse edge the fine links with 1 ≤ r ≤ B − 2 1\le r\le B-2 1 ≤ r ≤ B − 2 B B B ( x , μ ) (x,\mu) ( x , μ ) ν ≠ μ \nu\ne\mu ν = μ μ \mu μ r , r + 1 ∈ [ 1 , B − 1 ] r,r+1\in[1,B-1] r , r + 1 ∈ [ 1 , B − 1 ] B x Bx B x ν \nu ν B Z B\mathbb{Z} B Z e r i x ⋅ u i / Z ( r i ) e^{r_i x\cdot u_i}/Z(r_i) e r i x ⋅ u i / Z ( r i ) 0 ≤ r i ≤ 6 b 0\le r_i\le6b 0 ≤ r i ≤ 6 b l ≥ 1 l\ge1 l ≥ 1 D ( l ) ( u i ) D^{(l)}(u_i) D ( l ) ( u i ) I l + 1 ( r i ) / I 1 ( r i ) I_{l+1}(r_i)/I_1(r_i) I l + 1 ( r i ) / I 1 ( r i ) I 2 ( r i ) / I 1 ( r i ) I_2(r_i)/I_1(r_i) I 2 ( r i ) / I 1 ( r i ) I ν ( r ) I_\nu(r) I ν ( r ) ν ≥ 0 \nu\ge0 ν ≥ 0 I 2 ( r i ) / I 1 ( r i ) = E X ≤ 4 b / ( 4 b + 1 ) I_2(r_i)/I_1(r_i)=\mathbb{E}X\le4b/(4b+1) I 2 ( r i ) / I 1 ( r i ) = E X ≤ 4 b / ( 4 b + 1 ) L 2 L^2 L 2 e 6 b e^{6b} e 6 b Z ≥ 1 Z\ge1 Z ≥ 1 1 1 1 L 2 L^2 L 2 e 3 b e^{3b} e 3 b V e V_e V e B − 2 B-2 B − 2 m m m 1 2 e 3 b ( 4 b / ( 4 b + 1 ) ) B − 3 \frac12e^{3b}(4b/(4b+1))^{B-3} 2 1 e 3 b ( 4 b / ( 4 b + 1 ) ) B − 3 m ⊗ K m^{\otimes K} m ⊗ K F F F ω \omega ω m ⊗ K m^{\otimes K} m ⊗ K ω ( ϑ ( F ‾ ) F c ) \omega(\vartheta(\overline{F})F^c) ω ( ϑ ( F ) F c ) 2 ϵ 2\epsilon 2 ϵ ∣ ω ( F ) ∣ 2 |\omega(F)|^2 ∣ ω ( F ) ∣ 2 4 ϵ 4\epsilon 4 ϵ 4 ϵ 4\epsilon 4 ϵ
With B = ℓ / a k B=\ell/a_k B = ℓ / a k ℓ \ell ℓ b k = O ( log ( 1 / a k ) ) b_k=O(\log(1/a_k)) b k = O ( log ( 1/ a k )) Z k Z_k Z k Z k F c Z_kF^c Z k F c c > 0 c>0 c > 0 ( c / 6 ϵ ) 1 / 2 (c/6\epsilon)^{1/2} ( c /6 ϵ ) 1/2
What the cube estimate requires
This subsection proves seven facts about D ( n , α ) D(n,\alpha) D ( n , α ) n n n Theorem 71 and Proposition 73 for S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) n n n β \beta β β \beta β n n n β \beta β
For a finite V ⊂ A V\subset A V ⊂ A j ∈ A ∖ V j\in A\setminus V j ∈ A ∖ V b ( V , j ) b(V,j) b ( V , j ) V V V j j j k ∼ i k\sim i k ∼ i
Lemma 68. Suppose that C e e ′ ≤ c 0 C_{ee'}\le c_0 C e e ′ ≤ c 0 e , e ′ e,e' e , e ′ 12 c 0 < 1 12c_0<1 12 c 0 < 1 C V , j ≤ c 0 b ( V , j ) / ( 1 − 12 c 0 ) C_{V,j}\le c_0b(V,j)/(1-12c_0) C V , j ≤ c 0 b ( V , j ) / ( 1 − 12 c 0 ) V ⊂ A V\subset A V ⊂ A j ∈ A ∖ V j\in A\setminus V j ∈ A ∖ V
Proof. Fix Q , Q ′ Q,Q' Q , Q ′ j j j d = δ ( Q j , Q j ′ ) d=\delta(Q_j,Q'_j) d = δ ( Q j , Q j ′ ) μ = W ( ⋅ ∣ Q ) \mu=\mathcal{W}(\cdot\mid Q) μ = W ( ⋅ ∣ Q ) ν = W ( ⋅ ∣ Q ′ ) \nu=\mathcal{W}(\cdot\mid Q') ν = W ( ⋅ ∣ Q ′ ) μ \mu μ ν \nu ν π ↦ ∫ d γ d π \pi\mapsto\int d\gamma\,d\pi π ↦ ∫ d γ d π π \pi π ρ i = ∫ δ ( X i , Y i ) d π \rho_i=\int\delta(X_i,Y_i)\,d\pi ρ i = ∫ δ ( X i , Y i ) d π i ∈ V i\in V i ∈ V μ \mu μ X i X_i X i V V V γ i F ( ⋅ ∣ X ) \gamma_i^{\mathsf{F}}(\cdot\mid X) γ i F ( ⋅ ∣ X ) ν \nu ν ( X i , Y i ) (X_i,Y_i) ( X i , Y i ) γ i F ( ⋅ ∣ X ) \gamma_i^{\mathsf{F}}(\cdot\mid X) γ i F ( ⋅ ∣ X ) γ i F ( ⋅ ∣ Y ) \gamma_i^{\mathsf{F}}(\cdot\mid Y) γ i F ( ⋅ ∣ Y ) [75 Cor. 5.22] , and keep the other coordinates. The new coupling still has marginals μ \mu μ ν \nu ν X X X Y Y Y
W 1 ( γ i F ( ⋅ ∣ X ) , γ i F ( ⋅ ∣ Y ) ) ≤ ∑ k ∈ V k ≠ i C i k δ ( X k , Y k ) + C i j d . W_1\bigl(\gamma_i^{\mathsf{F}}(\cdot\mid X),\gamma_i^{\mathsf{F}}(\cdot\mid Y)\bigr)\leq\sum_{\substack{k\in V\\k\neq i}}C_{ik}\delta(X_k,Y_k)+C_{ij}d. W 1 ( γ i F ( ⋅ ∣ X ) , γ i F ( ⋅ ∣ Y ) ) ≤ k ∈ V k = i ∑ C ik δ ( X k , Y k ) + C ij d . Minimality of π \pi π ρ i ≤ c 0 ∑ k ∈ V k ∼ i ρ k + c 0 [ i ∼ j ] d \rho_i\leq c_0\sum_{\substack{k\in V\\k\sim i}}\rho_k+c_0[i\sim j]d ρ i ≤ c 0 ∑ k ∈ V k ∼ i ρ k + c 0 [ i ∼ j ] d i ∈ V i\in V i ∈ V ρ k \rho_k ρ k Lemma 9 (iv), so ( 1 − 12 c 0 ) ∑ i ρ i ≤ c 0 b ( V , j ) d (1-12c_0)\sum_i\rho_i\leq c_0 b(V,j)d ( 1 − 12 c 0 ) ∑ i ρ i ≤ c 0 b ( V , j ) d W 1 d V ( μ , ν ) ≤ ∑ i ρ i W_1^{\mathrm{dV}}(\mu,\nu)\leq\sum_i\rho_i W 1 dV ( μ , ν ) ≤ ∑ i ρ i d d d
Proposition 69. Let n = 2 m n=2m n = 2 m V V V Definition 42 . For every class c c c
∑ j ∈ c ∖ V b ( V , j ) ≤ 10 m 3 − 2 m 2 . \sum_{j\in c\setminus V} b(V,j)\leq10m^3-2m^2. j ∈ c ∖ V ∑ b ( V , j ) ≤ 10 m 3 − 2 m 2 . with equality for three classes. Consequently, if C e e ′ ≤ c 0 C_{ee'}\leq c_0 C e e ′ ≤ c 0 e , e ′ e,e' e , e ′ 12 c 0 < 1 12c_0<1 12 c 0 < 1 D ( n , α ) D(n,\alpha) D ( n , α ) α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 )
α ≥ α ^ n = c 0 1 − 12 c 0 ( 20 n − 8 n 2 ) . \alpha\geq\widehat{\alpha}_n=\frac{c_0}{1-12c_0}\left(\frac{20}{n}-\frac{8}{n^2}\right). α ≥ α n = 1 − 12 c 0 c 0 ( n 20 − n 2 8 ) . For S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) ∣ β W ∣ < 1 3 |\beta_W|<\frac{1}{3} ∣ β W ∣ < 3 1 α ^ n = ∣ β W ∣ 1 − 3 ∣ β W ∣ ( 5 n − 2 n 2 ) \widehat{\alpha}_n=\frac{|\beta_W|}{1-3|\beta_W|}\left(\frac{5}{n}-\frac{2}{n^2}\right) α n = 1 − 3∣ β W ∣ ∣ β W ∣ ( n 5 − n 2 2 ) n > 5 ∣ β W ∣ / ( 1 − 3 ∣ β W ∣ ) n>5|\beta_W|/(1-3|\beta_W|) n > 5∣ β W ∣/ ( 1 − 3∣ β W ∣ )
Proof. Let j = ( y , μ ) j=(y,\mu) j = ( y , μ ) c c c y ≡ r y\equiv r y ≡ r V V V j j j y + d y+d y + d d ∈ { − 1 , 0 , 1 } 4 d\in\{-1,0,1\}^4 d ∈ { − 1 , 0 , 1 } 4 d d d c c c y ≡ r y\equiv r y ≡ r y + d ∈ [ 0 , n ) 4 y+d\in[0,n)^4 y + d ∈ [ 0 , n ) 4 y ∉ [ 0 , n ) 4 y\notin[0,n)^4 y ∈ / [ 0 , n ) 4 l l l m m m y l ≡ r l y_l\equiv r_l y l ≡ r l y l + d l ∈ [ 0 , n ) y_l+d_l\in[0,n) y l + d l ∈ [ 0 , n ) [ 0 , n ) [0,n) [ 0 , n ) d l = 1 d_l=1 d l = 1 r l = 1 r_l=1 r l = 1 y l = − 1 y_l=-1 y l = − 1 d l = − 1 d_l=-1 d l = − 1 r l = 0 r_l=0 r l = 0 y l = n y_l=n y l = n q q q m 4 − m 4 − q ( m − 1 ) q m^4-m^{4-q}(m-1)^q m 4 − m 4 − q ( m − 1 ) q 0 0 0 m 3 m^3 m 3 2 m 3 − m 2 2m^3-m^2 2 m 3 − m 2 q = 0 , 1 , 2 q=0,1,2 q = 0 , 1 , 2 a c m 3 − b c m 2 a_cm^3-b_cm^2 a c m 3 − b c m 2 b c b_c b c q = 2 q=2 q = 2 a c a_c a c q = 1 q=1 q = 1 b c b_c b c Table III , the pairs ( a c , b c ) (a_c,b_c) ( a c , b c ) ( 2 , 0 ) (2,0) ( 2 , 0 ) ( 7 , 0 ) (7,0) ( 7 , 0 ) ( 7 , 1 ) (7,1) ( 7 , 1 ) ( 9 , 1 ) (9,1) ( 9 , 1 ) ( 9 , 2 ) (9,2) ( 9 , 2 ) ( 10 , 2 ) (10,2) ( 10 , 2 ) ( 3 , 0 ) (3,0) ( 3 , 0 ) ( 4 , 0 ) (4,0) ( 4 , 0 ) ( 5 , 0 ) (5,0) ( 5 , 0 ) ( 6 , 0 ) (6,0) ( 6 , 0 ) ( 8 , 1 ) (8,1) ( 8 , 1 ) a c ≤ 10 a_c\leq10 a c ≤ 10 8 − a c + b c ≥ 0 8-a_c+b_c\geq0 8 − a c + b c ≥ 0
( 10 m 3 − 2 m 2 ) − ( a c m 3 − b c m 2 ) = m 2 ( ( 10 − a c ) m + b c − 2 ) (10m^3-2m^2)-(a_cm^3-b_cm^2)=m^2\bigl((10-a_c)m+b_c-2\bigr) ( 10 m 3 − 2 m 2 ) − ( a c m 3 − b c m 2 ) = m 2 ( ( 10 − a c ) m + b c − 2 ) ≥ m 2 ( 8 − a c + b c ) ≥ 0. \geq m^2(8-a_c+b_c)\geq0. ≥ m 2 ( 8 − a c + b c ) ≥ 0. with equality for the three classes with ( a c , b c ) = ( 10 , 2 ) (a_c,b_c)=(10,2) ( a c , b c ) = ( 10 , 2 ) K c ≤ c 0 ( 10 m 3 − 2 m 2 ) / ( 1 − 12 c 0 ) K_c\leq c_0(10m^3-2m^2)/(1-12c_0) K c ≤ c 0 ( 10 m 3 − 2 m 2 ) / ( 1 − 12 c 0 ) N n = m 4 N_n=m^4 N n = m 4 K c / N n ≤ α ^ n K_c/N_n\leq\widehat{\alpha}_n K c / N n ≤ α n S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) Theorem 1 gives c 0 = ∣ β ∣ = ∣ β W ∣ / 4 c_0=|\beta|=|\beta_W|/4 c 0 = ∣ β ∣ = ∣ β W ∣/4
At β W = 3 10 \beta_W=\frac{3}{10} β W = 10 3 n = 16 n=16 n = 16 α ^ 16 = 117 128 \widehat{\alpha}_{16}=\frac{117}{128} α 16 = 128 117 Theorem 43 gives the gap log 128 117 / 32 \log\frac{128}{117}/32 log 117 128 /32 Theorem 1 .
The proposition shows that the cube condition is not empty; it does not extend the range, because the absorption 12 c 0 < 1 12c_0<1 12 c 0 < 1
The cube condition is also stable under small changes of the action, which is what a block renormalisation would need at its last scale, where the effective action is no longer of Wilson form. Let r ≥ 0 r\geq0 r ≥ 0 ε ≥ 0 \varepsilon\geq0 ε ≥ 0 ( w X ) (w_X) ( w X ) G A G^A G A X ⊂ A X\subset A X ⊂ A r r r w X w_X w X X X X 2 Z 4 2\mathbb{Z}^4 2 Z 4 M M M X X X X X X s s s
∣ w X ( Q ) − w X ( Q ′ ) ∣ ≤ ε ∑ i ∈ X δ ( Q i , Q i ′ ) \left|w_X(Q)-w_X(Q')\right|\leq\varepsilon\sum_{i\in X}\delta(Q_i,Q_i') ∣ w X ( Q ) − w X ( Q ′ ) ∣ ≤ ε i ∈ X ∑ δ ( Q i , Q i ′ ) for all Q , Q ′ Q,Q' Q , Q ′ γ V W ( ⋅ ∣ Q ) \gamma_V^W(\cdot\mid Q) γ V W ( ⋅ ∣ Q ) exp ∑ X ∩ V ≠ ∅ w X \exp\sum_{X\cap V\neq\varnothing}w_X exp ∑ X ∩ V = ∅ w X γ V ( ⋅ ∣ Q ) \gamma_V(\cdot\mid Q) γ V ( ⋅ ∣ Q ) C V , j W C_{V,j}^W C V , j W K c W K_c^W K c W D W ( n , α ) D^W(n,\alpha) D W ( n , α ) Definition 42 , with γ V W \gamma_V^W γ V W γ V \gamma_V γ V
Proposition 70. Let n n n D ( n , α ) D(n,\alpha) D ( n , α ) β \beta β D W ( n , α ′ ) D^W(n,\alpha') D W ( n , α ′ )
α ′ = α e Ω ( 1 + 2 Ω ) + 96 s Ω , Ω = 48 M D n ε . \alpha'=\alpha e^{\Omega}(1+2\Omega)+96s\Omega,\qquad\Omega=48MD_n\varepsilon. α ′ = α e Ω ( 1 + 2Ω ) + 96 s Ω , Ω = 48 M D n ε . In particular α ′ < 1 \alpha'<1 α ′ < 1 ε \varepsilon ε ε 0 ( n , α , M , s ) > 0 \varepsilon_0(n,\alpha,M,s)>0 ε 0 ( n , α , M , s ) > 0
Proof. The cube has ∣ V ∣ = 48 N n |V|=48N_n ∣ V ∣ = 48 N n d V d_V d V Δ V = 48 N n D \Delta_V=48N_nD Δ V = 48 N n D u = ∑ X ∩ V ≠ ∅ w X u=\sum_{X\cap V\neq\varnothing}w_X u = ∑ X ∩ V = ∅ w X G V G^V G V V V V M M M X X X u u u M ε M\varepsilon M ε d V d_V d V M ε Δ V = Ω M\varepsilon\Delta_V=\Omega M ε Δ V = Ω u u u sup u = 0 \sup u=0 sup u = 0 h = e u h=e^u h = e u e − Ω ≤ h ≤ 1 e^{-\Omega}\leq h\leq1 e − Ω ≤ h ≤ 1 h h h M ε M\varepsilon M ε
A common tilt. Let μ , ν \mu,\nu μ , ν G V G^V G V μ h = h μ / μ ( h ) \mu_h=h\mu/\mu(h) μ h = h μ / μ ( h ) ν h = h ν / ν ( h ) \nu_h=h\nu/\nu(h) ν h = h ν / ν ( h ) f f f 1 1 1 d V d_V d V ∣ f ∣ ≤ Δ V |f|\leq\Delta_V ∣ f ∣ ≤ Δ V f h fh f h ( 1 + Δ V M ε ) (1+\Delta_VM\varepsilon) ( 1 + Δ V M ε ) ∣ ν ( f h ) ∣ ≤ Δ V ν ( f ) |\nu(fh)|\leq\Delta_V\nu(f) ∣ ν ( f h ) ∣ ≤ Δ V ν ( f ) ∣ μ ( h ) − ν ( h ) ∣ ≤ M ε W 1 d V ( μ , ν ) |\mu(h)-\nu(h)|\leq M\varepsilon W_1^{\mathrm{dV}}(\mu,\nu) ∣ μ ( h ) − ν ( h ) ∣ ≤ M ε W 1 dV ( μ , ν ) μ ( h ) ≥ e − Ω \mu(h)\geq e^{-\Omega} μ ( h ) ≥ e − Ω
μ h ( f ) − ν h ( f ) = μ ( f h ) − ν ( f h ) μ ( h ) + ν ( f ) ν ( h ) − μ ( h ) μ ( h ) ν ( h ) ≤ e Ω ( 1 + 2 Ω ) W 1 d V ( μ , ν ) , \begin{aligned}
\mu_h(f)-\nu_h(f)=\frac{\mu(fh)-\nu(fh)}{\mu(h)}+\nu(f)\frac{\nu(h)-\mu(h)}{\mu(h)\nu(h)} \\
&\leq e^\Omega(1+2\Omega)W_1^{\mathrm{dV}}(\mu,\nu),
\end{aligned} μ h ( f ) − ν h ( f ) = μ ( h ) μ ( f h ) − ν ( f h ) + ν ( f ) μ ( h ) ν ( h ) ν ( h ) − μ ( h ) ≤ e Ω ( 1 + 2Ω ) W 1 dV ( μ , ν ) , since Δ V M ε = Ω \Delta_VM\varepsilon=\Omega Δ V M ε = Ω W 1 d V ( μ h , ν h ) ≤ e Ω ( 1 + 2 Ω ) W 1 d V ( μ , ν ) W_1^{\mathrm{dV}}(\mu_h,\nu_h)\leq e^\Omega(1+2\Omega)W_1^{\mathrm{dV}}(\mu,\nu) W 1 dV ( μ h , ν h ) ≤ e Ω ( 1 + 2Ω ) W 1 dV ( μ , ν )
Two tilts. If u ′ u' u ′ ∣ u − u ′ ∣ ≤ η |u-u'|\leq\eta ∣ u − u ′ ∣ ≤ η ν h \nu_h ν h ν h ′ \nu_{h'} ν h ′ [ e − 2 η , e 2 η ] [e^{-2\eta},e^{2\eta}] [ e − 2 η , e 2 η ] 1 − e − 2 η ≤ 2 η 1-e^{-2\eta}\leq2\eta 1 − e − 2 η ≤ 2 η W 1 d V ( ν h , ν h ′ ) ≤ 2 η Δ V W_1^{\mathrm{dV}}(\nu_h,\nu_{h'})\leq2\eta\Delta_V W 1 dV ( ν h , ν h ′ ) ≤ 2 η Δ V
The coefficients. Let Q , Q ′ Q,Q' Q , Q ′ j ∉ V j\notin V j ∈ / V δ = δ ( Q j , Q j ′ ) \delta=\delta(Q_j,Q'_j) δ = δ ( Q j , Q j ′ ) u , u ′ u,u' u , u ′ X X X j j j V V V m j m_j m j ∣ u − u ′ ∣ ≤ m j ε δ |u-u'|\leq m_j\varepsilon\delta ∣ u − u ′ ∣ ≤ m j ε δ μ = γ V ( ⋅ ∣ Q ) \mu=\gamma_V(\cdot\mid Q) μ = γ V ( ⋅ ∣ Q ) ν = γ V ( ⋅ ∣ Q ′ ) \nu=\gamma_V(\cdot\mid Q') ν = γ V ( ⋅ ∣ Q ′ )
C V , j W ≤ e Ω ( 1 + 2 Ω ) C V , j + 2 Δ V m j ε . C_{V,j}^W\leq e^\Omega(1+2\Omega)C_{V,j}+2\Delta_Vm_j\varepsilon. C V , j W ≤ e Ω ( 1 + 2Ω ) C V , j + 2 Δ V m j ε . The pairs ( X , j ) (X,j) ( X , j ) j ∈ X ∖ V j\in X\setminus V j ∈ X ∖ V X ∩ V ≠ ∅ X\cap V\ne\varnothing X ∩ V = ∅ s s s V V V s M ∣ V ∣ sM|V| s M ∣ V ∣ j ∈ c ∖ V j\in c\setminus V j ∈ c ∖ V
K c W ≤ e Ω ( 1 + 2 Ω ) α N n + 2 Δ V s M ∣ V ∣ ε = ( α ε Ω ( 1 + 2 Ω ) + 96 s Ω ) N n . \begin{aligned}
K_c^W\le e^{\Omega(1+2\Omega)}\alpha N_n+2\Delta_VsM|V|\varepsilon\\
&=(\alpha\varepsilon^{\Omega}(1+2\Omega)+96s\Omega)N_n.
\end{aligned} K c W ≤ e Ω ( 1 + 2Ω ) α N n + 2 Δ V s M ∣ V ∣ ε = ( α ε Ω ( 1 + 2Ω ) + 96 s Ω ) N n . The family is invariant under 2 Z 4 2\mathbb{Z}^4 2 Z 4 D W ( n , α ′ ) D^W(n,\alpha') D W ( n , α ′ )
The admissible perturbation is of order 1 / N n = ( 2 / n ) 4 1/N_n=(2/n)^4 1/ N n = ( 2/ n ) 4 ℓ \ell ℓ ε 0 \varepsilon_0 ε 0
The next theorem uses the configuration in which every link outside V V V γ V 1 = γ V ( ⋅ ∣ 1 ) \gamma_V^1=\gamma_V(\mathord\cdot\mid1) γ V 1 = γ V ( ⋅ ∣ 1 )
ε V ( β ) = 1 ∣ V ∣ ∑ i ∈ V ∫ δ ( 1 , Q i ) d γ V 1 ( Q ) \varepsilon_V(\beta)=\frac{1}{|V|}\sum_{i\in V}\int\delta(1,Q_i)\,d\gamma_V^1(Q) ε V ( β ) = ∣ V ∣ 1 i ∈ V ∑ ∫ δ ( 1 , Q i ) d γ V 1 ( Q ) for the mean distance of the gauge-fixed variables of the cube from the identity. Recall that δ \delta δ D D D
Theorem 71. Let n n n V V V Definition 42 , and suppose that D ( n , α ) D(n,\alpha) D ( n , α ) β \beta β
ε V ( β ) ≥ ( 1 − α ) D 3 . \varepsilon_V(\beta)\ge\frac{(1-\alpha)D}{3}. ε V ( β ) ≥ 3 ( 1 − α ) D . Proof. A flat configuration. By Lemma 9 (i) and the period two of σ \sigma σ { r , r + e σ ( r ) } \{r,r+e_{\sigma(r)}\} { r , r + e σ ( r ) } F F F z , w ∈ G z,w\in G z , w ∈ G δ ( z , w ) = D \delta(z,w)=D δ ( z , w ) = D g ( x ) = w g(x)=w g ( x ) = w x x x g ( x ) = z g(x)=z g ( x ) = z Q ( x , μ ) g = g ( x ) g ( x + e μ ) − 1 Q^g_{(x,\mu)}=g(x)g(x+e_\mu)^{-1} Q ( x , μ ) g = g ( x ) g ( x + e μ ) − 1 Z 4 \mathbb{Z}^4 Z 4 g g g F F F Q g Q^g Q g 1 1 1 F F F Q g Q^g Q g 1 1 1 Q g Q^g Q g δ ( 1 , Q ( x , μ ) g ) = δ ( g ( x + e μ ) , g ( x ) ) \delta(1,Q^g_{(x,\mu)})=\delta(g(x+e_\mu),g(x)) δ ( 1 , Q ( x , μ ) g ) = δ ( g ( x + e μ ) , g ( x )) D D D x x x x + e μ x+e_\mu x + e μ 0 0 0 tbl:III the first case occurs for exactly 32 of the 48 classes. Let δ c \delta_c δ c δ ( 1 , Q e g ) \delta(1,Q^g_e) δ ( 1 , Q e g ) e e e c c c ∑ c δ c = 32 D \sum_c\delta_c=32D ∑ c δ c = 32 D
Gauge covariance. Let Φ ( Q ) ( x , μ ) = g ( x ) Q ( x , μ ) g ( x + e μ ) − 1 \Phi(Q)_{(x,\mu)}=g(x)Q_{(x,\mu)}g(x+e_\mu)^{-1} Φ ( Q ) ( x , μ ) = g ( x ) Q ( x , μ ) g ( x + e μ ) − 1 G V G^{V} G V g g g Q V Q_V Q V V V V 1 1 1 Φ ( Q V ) \Phi(Q_V) Φ ( Q V ) V V V Q g Q^g Q g F F F 1 1 1 Φ \Phi Φ γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V 1 \gamma_V(\mathord\cdot\mid Q^g)=\Phi_*\gamma_V^1 γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V 1
Upper bound. Only the finitely many links outside V V V V V V γ V \gamma_V γ V 1 1 1 Q g Q^g Q g C V , j C_{V,j} C V , j
W 1 d v ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≤ ∑ j ∈ A ∖ V C V , j δ ( 1 , Q j g ) = ∑ c δ c K c ≤ α N n ∑ c δ c = 32 α D N n . \begin{aligned}
W_1^{\mathrm{dv}}(\gamma_V^1,\gamma_V(\mathord\cdot\mid Q^g))\le\sum_{j\in A\setminus V}C_{V,j}\,\delta(1,Q^g_j)=\sum_c\delta_cK_c \\
&\le\alpha N_n\sum_c\delta_c=32\alpha DN_n.
\end{aligned} W 1 dv ( γ V 1 , γ V ( ⋅ ∣ Q g )) ≤ j ∈ A ∖ V ∑ C V , j δ ( 1 , Q j g ) = c ∑ δ c K c ≤ α N n c ∑ δ c = 32 α D N n . Lower bound. The function f ( Q ) = − ∑ i ∈ V δ ( 1 , Q i g ) f(Q)=-\sum_{i\in V}\delta(1,Q_i^g) f ( Q ) = − ∑ i ∈ V δ ( 1 , Q i g ) 1 1 1 d v \mathrm{dv} dv ∫ f d γ V ( ⋅ ∣ Q g ) = − ∫ ∑ i δ ( 1 , Q i ) d γ V 1 = − ∣ V ∣ ε V \int f\,d\gamma_V(\mathord\cdot\mid Q^g)=-\int\sum_i\delta(1,Q_i)\,d\gamma_V^1=-|V|\varepsilon_V ∫ f d γ V ( ⋅ ∣ Q g ) = − ∫ ∑ i δ ( 1 , Q i ) d γ V 1 = − ∣ V ∣ ε V ∫ f d γ V 1 ≤ − ∑ i δ ( 1 , Q i g ) + ∣ V ∣ ε V \int f\,d\gamma_V^1\le-\sum_i\delta(1,Q_i^g)+|V|\varepsilon_V ∫ f d γ V 1 ≤ − ∑ i δ ( 1 , Q i g ) + ∣ V ∣ ε V N n N_n N n V V V ∑ i δ ( 1 , Q i g ) = 32 D N n \sum_i\delta(1,Q_i^g)=32DN_n ∑ i δ ( 1 , Q i g ) = 32 D N n
W 1 d v ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≥ 32 D N n − 2 ∣ V ∣ ε V . W_1^{\mathrm{dv}}(\gamma_V^1,\gamma_V(\mathord\cdot\mid Q^g))\ge32DN_n-2|V|\varepsilon_V. W 1 dv ( γ V 1 , γ V ( ⋅ ∣ Q g )) ≥ 32 D N n − 2∣ V ∣ ε V . With ∣ V ∣ = 48 N n |V|=48N_n ∣ V ∣ = 48 N n 96 N n ε V ≥ 32 ( 1 − α ) D N n 96N_n\varepsilon_V\ge32(1-\alpha)DN_n 96 N n ε V ≥ 32 ( 1 − α ) D N n
Corollary 72. For every even n n n α < 1 \alpha<1 α < 1 β 0 ( n , α ) \beta_0(n,\alpha) β 0 ( n , α ) D ( n , α ) D(n,\alpha) D ( n , α ) β ≥ β 0 ( n , α ) \beta\ge\beta_0(n,\alpha) β ≥ β 0 ( n , α ) D ( n k , α ) D(n_k,\alpha) D ( n k , α ) β k → ∞ \beta_k\to\infty β k → ∞ n k → ∞ n_k\to\infty n k → ∞
Proof. For β > 0 \beta>0 β > 0 γ V 1 \gamma_V^1 γ V 1 e − N β S e^{-N\beta S} e − N β S S = ∑ p ∩ V ≠ ∅ ( N − Re Tr Q p F ) S=\sum_{p\cap V\ne\varnothing}(N-\operatorname{Re}\operatorname{Tr}Q_p^F) S = ∑ p ∩ V = ∅ ( N − Re Tr Q p F ) G V G^V G V Q p F = 1 Q_p^F=1 Q p F = 1 p p p V V V Q V = 1 Q_V=1 Q V = 1 1 1 1 1 1 1 Z 4 \mathbb{Z}^4 Z 4 h h h Q ( x , μ ) = h ( x ) h ( x + e μ ) − 1 Q_{(x,\mu)}=h(x)h(x+e_\mu)^{-1} Q ( x , μ ) = h ( x ) h ( x + e μ ) − 1 h h h Q = 1 Q=1 Q = 1 [ 0 , n ) 4 [0,n)^4 [ 0 , n ) 4 x ν x_\nu x ν ν \nu ν n n n h h h F F F h h h h h h Q V = 1 Q_V=1 Q V = 1
Let U U U 1 1 1 G V G^V G V s = min G V ∖ U S > 0 s=\min_{G^V\setminus U}S>0 s = min G V ∖ U S > 0 U ′ ⊂ U U'\subset U U ′ ⊂ U 1 1 1 S < s / 2 S<s/2 S < s /2 γ V 1 ( G V ∖ U ) ≤ e − N β s ∣ U ′ ∣ / e − N β s / 2 \gamma_V^1(G^V\setminus U)\le e^{-N\beta s}|U'|/e^{-N\beta s/2} γ V 1 ( G V ∖ U ) ≤ e − N β s ∣ U ′ ∣/ e − N β s /2 ε V ( β ) → 0 \varepsilon_V(\beta)\to0 ε V ( β ) → 0 β → ∞ \beta\to\infty β → ∞ Theorem 71 excludes D ( n , α ) D(n,\alpha) D ( n , α ) ε V ( β ) < ( 1 − α ) D / 3 \varepsilon_V(\beta)<(1-\alpha)D/3 ε V ( β ) < ( 1 − α ) D /3
For SU(2) the constant of Theorem 71 can be improved: a flat configuration with four values instead of two moves every class of active links, and the means of the gauge-fixed links are scalar.
Proposition 73. Let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) n n n V V V Definition 42 , and suppose that D ( n , α ) D(n,\alpha) D ( n , α ) β \beta β
1 ∣ V ∣ ∑ i ∈ V ∣ ∫ Re Tr Q i d γ V 1 ∣ ≤ 2 α , \frac{1}{|V|}\sum_{i\in V}\left|\int\operatorname{Re}\operatorname{Tr}Q_i\,d\gamma_V^1\right|\le2\alpha, ∣ V ∣ 1 i ∈ V ∑ ∫ Re Tr Q i d γ V 1 ≤ 2 α , 1 ∣ V ∣ ∑ i ∈ V ∫ δ ( 1 , Q i ) 2 d γ V 1 ≥ 2 ( 1 − α ) , \frac{1}{|V|}\sum_{i\in V}\int\delta(1,Q_i)^2\,d\gamma_V^1\ge2(1-\alpha), ∣ V ∣ 1 i ∈ V ∑ ∫ δ ( 1 , Q i ) 2 d γ V 1 ≥ 2 ( 1 − α ) , and ε V ( β ) ≥ 1 − α \varepsilon_V(\beta)\ge1-\alpha ε V ( β ) ≥ 1 − α
Proof. Write ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ Re Tr q = 2 ⟨ q , 1 ⟩ \operatorname{Re}\operatorname{Tr}q=2\langle q,1\rangle Re Tr q = 2 ⟨ q , 1 ⟩ δ ( p , q ) = ∣ p − q ∣ \delta(p,q)=|p-q| δ ( p , q ) = ∣ p − q ∣
Four colours. Split the sixteen residues modulo two into the sets
X 1 = { 0000 , 1000 , 0111 , 1111 } , X_1=\{0000,1000,0111,1111\}, X 1 = { 0000 , 1000 , 0111 , 1111 } , X 2 = { 0001 , 0101 , 1010 , 1110 } , X_2=\{0001,0101,1010,1110\}, X 2 = { 0001 , 0101 , 1010 , 1110 } , X 3 = { 0010 , 0011 , 1100 , 1101 } , X_3=\{0010,0011,1100,1101\}, X 3 = { 0010 , 0011 , 1100 , 1101 } , X 4 = { 0100 , 0110 , 1001 , 1011 } . X_4=\{0100,0110,1001,1011\}. X 4 = { 0100 , 0110 , 1001 , 1011 } . Each is the union of two of the eight pairs { r , r + e σ ( r ) } \{r,r+e_{\sigma(r)}\} { r , r + e σ ( r ) } Table III shows that r r r r + e μ r+e_\mu r + e μ r r r μ ≠ σ ( r ) \mu\ne\sigma(r) μ = σ ( r ) Theorem 71 form X 3 ∪ X 4 X_3\cup X_4 X 3 ∪ X 4
z 1 = ( i + j + k ) / 3 , z 2 = ( i − j − k ) / 3 , z_1=(\mathrm{i}+\mathrm{j}+\mathrm{k})/\sqrt{3},\qquad z_2=(\mathrm{i}-\mathrm{j}-\mathrm{k})/\sqrt{3}, z 1 = ( i + j + k ) / 3 , z 2 = ( i − j − k ) / 3 , z 3 = ( − i + j − k ) / 3 , z 4 = ( − i − j + k ) / 3 , z_3=(-\mathrm{i}+\mathrm{j}-\mathrm{k})/\sqrt{3},\qquad z_4=(-\mathrm{i}-\mathrm{j}+\mathrm{k})/\sqrt{3}, z 3 = ( − i + j − k ) / 3 , z 4 = ( − i − j + k ) / 3 , unit quaternions with ⟨ z k , z l ⟩ = − 1 3 \langle z_k,z_l\rangle=-\frac{1}{3} ⟨ z k , z l ⟩ = − 3 1 k ≠ l k\ne l k = l δ ( z k , z l ) = d ∗ = ( 8 / 3 ) 1 / 2 \delta(z_k,z_l)=d_*=(8/3)^{1/2} δ ( z k , z l ) = d ∗ = ( 8/3 ) 1/2 g ( x ) = z k g(x)=z_k g ( x ) = z k x x x X k X_k X k Q ( x , μ ) g = g ( x ) g ( x + e μ ) − 1 Q^g_{(x,\mu)}=g(x)g(x+e_\mu)^{-1} Q ( x , μ ) g = g ( x ) g ( x + e μ ) − 1 Theorem 71 , Q g Q^g Q g 1 1 1 F F F Q g Q^g Q g 1 1 1 γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V 1 \gamma_V(\mathord{\cdot}\mid Q^g)=\Phi_*\gamma_V^1 γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V 1 Φ ( Q ) ( x , μ ) = g ( x ) Q ( x , μ ) g ( x + e μ ) − 1 \Phi(Q)_{(x,\mu)}=g(x)Q_{(x,\mu)}g(x+e_\mu)^{-1} Φ ( Q ) ( x , μ ) = g ( x ) Q ( x , μ ) g ( x + e μ ) − 1 δ ( 1 , Q e g ) = d ∗ \delta(1,Q^g_e)=d_* δ ( 1 , Q e g ) = d ∗ e e e
Upper bound. Changing the exterior links one at a time,
W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≤ d ∗ ∑ c K c ≤ 48 α d ∗ N n = α d ∗ ∣ V ∣ . W_1^{d_V}\bigl(\gamma_V^1,\gamma_V(\mathord{\cdot}\mid Q^g)\bigr)\le d_*\sum_c K_c\le48\alpha d_*N_n=\alpha d_*|V|. W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≤ d ∗ c ∑ K c ≤ 48 α d ∗ N n = α d ∗ ∣ V ∣. Lower bound. Conjugating every link of V V V 1 1 1 F F F γ V 1 \gamma_V^1 γ V 1 m i = ∫ Q i d γ V 1 m_i=\int Q_i\,d\gamma_V^1 m i = ∫ Q i d γ V 1 m i = a i 1 m_i=a_i1 m i = a i 1
a i = 1 2 ∫ Re Tr Q i d γ V 1 a_i=\frac{1}{2}\int\operatorname{Re}\operatorname{Tr}Q_i\,d\gamma_V^1 a i = 2 1 ∫ Re Tr Q i d γ V 1 real. Since Φ \Phi Φ Q i Q_i Q i γ V ( ⋅ ∣ Q g ) \gamma_V(\mathord{\cdot}\mid Q^g) γ V ( ⋅ ∣ Q g ) a i Q i g a_iQ_i^g a i Q i g w i w_i w i ⟨ w i , a i Q i g − a i 1 ⟩ = ∣ a i ∣ ∣ Q i g − 1 ∣ \langle w_i,a_iQ_i^g-a_i1\rangle=|a_i||Q_i^g-1| ⟨ w i , a i Q i g − a i 1 ⟩ = ∣ a i ∣∣ Q i g − 1∣ f ( Q ) = ∑ i ∈ V ⟨ w i , Q i ⟩ f(Q)=\sum_{i\in V}\langle w_i,Q_i\rangle f ( Q ) = ∑ i ∈ V ⟨ w i , Q i ⟩ 1 1 1 d V d_V d V
W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≥ ∑ i ∈ V ∣ a i ∣ δ ( 1 , Q i g ) = d ∗ ∑ i ∈ V ∣ a i ∣ . W_1^{d_V}\bigl(\gamma_V^1,\gamma_V(\mathord{\cdot}\mid Q^g)\bigr)\ge\sum_{i\in V}|a_i|\delta(1,Q_i^g)=d_*\sum_{i\in V}|a_i|. W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≥ i ∈ V ∑ ∣ a i ∣ δ ( 1 , Q i g ) = d ∗ i ∈ V ∑ ∣ a i ∣. Conclusion. Comparing the two bounds, ∑ i ∣ a i ∣ ≤ α ∣ V ∣ \sum_i|a_i|\le\alpha|V| ∑ i ∣ a i ∣ ≤ α ∣ V ∣ δ ( 1 , q ) 2 = 2 − 2 ⟨ q , 1 ⟩ \delta(1,q)^2=2-2\langle q,1\rangle δ ( 1 , q ) 2 = 2 − 2 ⟨ q , 1 ⟩
∫ δ ( 1 , Q i ) 2 d γ V 1 = 2 ( 1 − a i ) ≥ 2 ( 1 − ∣ a i ∣ ) , \int\delta(1,Q_i)^2\,d\gamma_V^1=2(1-a_i)\ge2(1-|a_i|), ∫ δ ( 1 , Q i ) 2 d γ V 1 = 2 ( 1 − a i ) ≥ 2 ( 1 − ∣ a i ∣ ) , and averaging gives the second. As δ ≤ 2 \delta\le2 δ ≤ 2 δ ≥ δ 2 / 2 \delta\ge\delta^2/2 δ ≥ δ 2 /2 ε V \varepsilon_V ε V
The four colours need nothing from SU(2) except four elements whose pairwise quotients are conjugate, and every compact simple group contains such elements. For a metric that comes from a representation the bound then compares the cube with Haar measure.
Proposition 74. Let G G G ρ \rho ρ G G G d d d δ ( p , q ) = κ − 1 / 2 ∥ ρ ( p ) − ρ ( q ) ∥ H S \delta(p,q)=\kappa^{-1/2}\lVert\rho(p)-\rho(q)\rVert_{\mathrm{HS}} δ ( p , q ) = κ − 1/2 ∥ ρ ( p ) − ρ ( q ) ∥ HS κ > 0 \kappa>0 κ > 0 n n n V V V D ( n , α ) D(n,\alpha) D ( n , α ) β \beta β
1 ∣ V ∣ ∑ i ∈ V ∫ ∣ Tr ρ ( Q i ) ∣ d γ V 1 ≤ α d , \frac{1}{|V|}\sum_{i\in V}\int\left|\operatorname{Tr}\rho(Q_i)\right|\,d\gamma_V^1\le\alpha d, ∣ V ∣ 1 i ∈ V ∑ ∫ ∣ Tr ρ ( Q i ) ∣ d γ V 1 ≤ α d , 1 ∣ V ∣ ∑ i ∈ V ∫ δ ( 1 , Q i ) 2 d γ V 1 ≥ ( 1 − α ) ∫ G δ ( 1 , q ) 2 d q , \frac{1}{|V|}\sum_{i\in V}\int\delta(1,Q_i)^2\,d\gamma_V^1\ge(1-\alpha)\int_G\delta(1,q)^2\,dq, ∣ V ∣ 1 i ∈ V ∑ ∫ δ ( 1 , Q i ) 2 d γ V 1 ≥ ( 1 − α ) ∫ G δ ( 1 , q ) 2 d q , where d q dq d q ε V ( β ) ≥ ( 1 − α ) D / 2 \varepsilon_V(\beta)\ge(1-\alpha)D/2 ε V ( β ) ≥ ( 1 − α ) D /2
Proof. Four elements. Let φ : SU ( 2 ) → G \varphi:\operatorname{SU}(2)\to G φ : SU ( 2 ) → G z 1 , … , z 4 z_1,\ldots,z_4 z 1 , … , z 4 k ≠ l k\ne l k = l z k z l − 1 z_kz_l^{-1} z k z l − 1 ⟨ z k , z l ⟩ = − 1 3 \langle z_k,z_l\rangle=-\frac{1}{3} ⟨ z k , z l ⟩ = − 3 1 φ ( z k z l − 1 ) \varphi(z_kz_l^{-1}) φ ( z k z l − 1 ) G G G δ ( φ ( z k ) , φ ( z l ) ) = d ∗ \delta(\varphi(z_k),\varphi(z_l))=d_* δ ( φ ( z k ) , φ ( z l )) = d ∗ k ≠ l k\ne l k = l d ∗ d_* d ∗ d ∗ = 0 d_*=0 d ∗ = 0 φ ( z k z l − 1 ) \varphi(z_kz_l^{-1}) φ ( z k z l − 1 ) ker ρ \ker\rho ker ρ G G G φ ( SU ( 2 ) ) \varphi(\operatorname{SU}(2)) φ ( SU ( 2 )) z k z l − 1 z_kz_l^{-1} z k z l − 1 { ± 1 } \{\pm1\} { ± 1 } ker φ ⊂ { ± 1 } \ker\varphi\subset\{\pm1\} ker φ ⊂ { ± 1 } − 1 3 -\frac{1}{3} − 3 1 d ∗ > 0 d_*>0 d ∗ > 0
The two bounds. Put g ( x ) = φ ( z k ) g(x)=\varphi(z_k) g ( x ) = φ ( z k ) x x x X k X_k X k Q g Q^g Q g 1 1 1 F F F δ ( 1 , Q e g ) = d ∗ \delta(1,Q_e^g)=d_* δ ( 1 , Q e g ) = d ∗ W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g ) ) ≤ α d ∗ ∣ V ∣ W_1^{d_V}(\gamma_V^1,\gamma_V(\mathord{\cdot}\mid Q^g))\le\alpha d_*|V| W 1 d V ( γ V 1 , γ V ( ⋅ ∣ Q g )) ≤ α d ∗ ∣ V ∣ V V V γ V 1 \gamma_V^1 γ V 1 ∫ ρ ( Q i ) d γ V 1 \int\rho(Q_i)\,d\gamma_V^1 ∫ ρ ( Q i ) d γ V 1 ρ ( G ) \rho(G) ρ ( G ) c i I c_iI c i I c i = d − 1 ∫ Tr ρ ( Q i ) d γ V 1 c_i=d^{-1}\int\operatorname{Tr}\rho(Q_i)\,d\gamma_V^1 c i = d − 1 ∫ Tr ρ ( Q i ) d γ V 1 γ V ( ⋅ ∣ Q g ) \gamma_V(\mathord{\cdot}\mid Q^g) γ V ( ⋅ ∣ Q g ) c i ρ ( Q i g ) c_i\rho(Q_i^g) c i ρ ( Q i g ) W i W_i W i κ − 1 / 2 \kappa^{-1/2} κ − 1/2 Re Tr ( W i ∗ c i ( ρ ( Q i g ) − I ) ) = ∣ c i ∣ δ ( 1 , Q i g ) \operatorname{Re}\operatorname{Tr}(W_i^*c_i(\rho(Q_i^g)-I))=|c_i|\delta(1,Q_i^g) Re Tr ( W i ∗ c i ( ρ ( Q i g ) − I )) = ∣ c i ∣ δ ( 1 , Q i g ) f ( Q ) = ∑ i Re Tr ( W i ∗ ρ ( Q i ) ) f(Q)=\sum_i\operatorname{Re}\operatorname{Tr}(W_i^*\rho(Q_i)) f ( Q ) = ∑ i Re Tr ( W i ∗ ρ ( Q i )) 1 1 1 d V d_V d V W 1 d V ≥ d ∗ ∑ i ∣ c i ∣ W_1^{d_V}\ge d_*\sum_i|c_i| W 1 d V ≥ d ∗ ∑ i ∣ c i ∣ ∑ i ∣ c i ∣ ≤ α ∣ V ∣ \sum_i|c_i|\le\alpha|V| ∑ i ∣ c i ∣ ≤ α ∣ V ∣
Comparison with Haar measure. Since κ δ ( 1 , q ) 2 = 2 d − 2 Re Tr ρ ( q ) \kappa\delta(1,q)^2=2d-2\operatorname{Re}\operatorname{Tr}\rho(q) κ δ ( 1 , q ) 2 = 2 d − 2 Re Tr ρ ( q ) ∫ G Tr ρ ( q ) d q = 0 \int_G\operatorname{Tr}\rho(q)\,dq=0 ∫ G Tr ρ ( q ) d q = 0 ρ \rho ρ ∫ G δ ( 1 , q ) 2 d q = 2 d / κ \int_G\delta(1,q)^2\,dq=2d/\kappa ∫ G δ ( 1 , q ) 2 d q = 2 d / κ ∫ δ ( 1 , Q i ) 2 d γ V 1 = 2 d ( 1 − Re c i ) / κ ≥ 2 d ( 1 − ∣ c i ∣ ) / κ \int\delta(1,Q_i)^2\,d\gamma_V^1=2d(1-\operatorname{Re}c_i)/\kappa\ge2d(1-|c_i|)/\kappa ∫ δ ( 1 , Q i ) 2 d γ V 1 = 2 d ( 1 − Re c i ) / κ ≥ 2 d ( 1 − ∣ c i ∣ ) / κ κ D 2 ≤ 4 d \kappa D^2\le4d κ D 2 ≤ 4 d ∥ ρ ( p ) − ρ ( q ) ∥ H S ≤ 2 d \lVert\rho(p)-\rho(q)\rVert_{\mathrm{HS}}\le2\sqrt d ∥ ρ ( p ) − ρ ( q ) ∥ HS ≤ 2 d δ ≥ δ 2 / D \delta\ge\delta^2/D δ ≥ δ 2 / D ε V ≥ 2 d ( 1 − α ) / ( κ D ) ≥ ( 1 − α ) D / 2 \varepsilon_V\ge2d(1-\alpha)/(\kappa D)\ge(1-\alpha)D/2 ε V ≥ 2 d ( 1 − α ) / ( κ D ) ≥ ( 1 − α ) D /2
For SU(N N N 1 3 \frac{1}{3} 3 1 Theorem 71 to 1 2 \frac{1}{2} 2 1 D 2 = 9 D^2=9 D 2 = 9 δ 2 \delta^2 δ 2 6 6 6 2 3 \frac{2}{3} 3 2 D ( n , α ) D(n,\alpha) D ( n , α ) 1 − α 1-\alpha 1 − α
The size β 0 ( n , α ) \beta_0(n,\alpha) β 0 ( n , α )
Proposition 75. Let G = SU ( 2 ) G=\operatorname{SU}(2) G = SU ( 2 ) D = 2 D=2 D = 2 b = β W ≥ 1 9 b=\beta_W\ge\frac{1}{9} b = β W ≥ 9 1 n ≥ 2 n\ge2 n ≥ 2 L ( b ) = 1 + log 24 + 3 2 log ( 9 b ) L(b)=1+\log24+\frac{3}{2}\log(9b) L ( b ) = 1 + log 24 + 2 3 log ( 9 b )
1 ∣ V ∣ ∑ e ∈ V ∫ δ ( 1 , Q e ) 2 d γ V 1 ≤ n 2 ( n + 2 ) ( 2 n + 1 ) L ( b ) b , (18) \frac{1}{|V|}\sum_{e\in V}\int\delta(1,Q_e)^2\,d\gamma_V^1\le\frac{n^2(n+2)(2n+1)L(b)}{b}, \tag*{(18)} ∣ V ∣ 1 e ∈ V ∑ ∫ δ ( 1 , Q e ) 2 d γ V 1 ≤ b n 2 ( n + 2 ) ( 2 n + 1 ) L ( b ) , (18) so ε V ( β ) 2 \varepsilon_V(\beta)^2 ε V ( β ) 2 D ( n , α ) D(n,\alpha) D ( n , α ) b b b
n 4 ≥ 2 ( 1 − α ) b 5 L ( b ) . n^4 \ge\frac{2(1-\alpha)b}{5L(b)}. n 4 ≥ 5 L ( b ) 2 ( 1 − α ) b . Proof. Put m = n / 2 m=n/2 m = n /2 γ V 1 \gamma_V^1 γ V 1 V V V F F F V V V
Links as loops. Let e = ( x , μ ) ∈ V e=(x,\mu)\in V e = ( x , μ ) ∈ V y = x + e μ y=x+e_\mu y = x + e μ σ = σ ( x ) \sigma=\sigma(x) σ = σ ( x ) τ = σ ( y ) \tau=\sigma(y) τ = σ ( y ) σ , τ , μ \sigma,\tau,\mu σ , τ , μ σ ≠ μ \sigma\ne\mu σ = μ e e e τ ≠ μ \tau\ne\mu τ = μ F F F y y y x x x F F F τ ≠ σ \tau\ne\sigma τ = σ Table III , in which σ ( x + e μ ) ≠ σ ( x ) \sigma(x+e_\mu)\ne\sigma(x) σ ( x + e μ ) = σ ( x ) μ ≠ σ ( x ) \mu\ne\sigma(x) μ = σ ( x ) 0 ≤ x σ < n 0\le x_\sigma<n 0 ≤ x σ < n n − x σ n-x_\sigma n − x σ x σ + 1 x_\sigma+1 x σ + 1 m m m s s s ε = ± 1 \varepsilon=\pm1 ε = ± 1 x ′ = x + s ε e σ x'=x+s\varepsilon e_\sigma x ′ = x + s ε e σ σ \sigma σ [ 0 , n ) [0,n) [ 0 , n ) t ≤ m t\le m t ≤ m ε ′ \varepsilon' ε ′ y y y τ \tau τ y τ = x τ y_\tau=x_\tau y τ = x τ R 1 R_1 R 1 s s s x x x x ′ + e μ x'+e_\mu x ′ + e μ y y y { σ , μ } \{\sigma,\mu\} { σ , μ } R 2 R_2 R 2 s t st s t y y y x ′ + e μ x'+e_\mu x ′ + e μ x ′ + e μ + t ε ′ e τ x'+e_\mu+t\varepsilon'e_\tau x ′ + e μ + t ε ′ e τ y + t ε ′ e τ y+t\varepsilon'e_\tau y + t ε ′ e τ { σ , τ } \{\sigma,\tau\} { σ , τ } R ( e ) = R 1 ∪ R 2 R(e)=R_1\cup R_2 R ( e ) = R 1 ∪ R 2 x x x F F F x x x x ′ x' x ′ x ′ + e μ x'+e_\mu x ′ + e μ e τ e_\tau e τ x ′ + e μ + t ε ′ e τ x'+e_\mu+t\varepsilon'e_\tau x ′ + e μ + t ε ′ e τ e σ e_\sigma e σ y + t ε ′ e τ y+t\varepsilon'e_\tau y + t ε ′ e τ F F F y y y e e e F F F σ \sigma σ x σ ′ ∉ [ 0 , n ) x'_\sigma\notin[0,n) x σ ′ ∈ / [ 0 , n ) τ \tau τ [ 0 , n ) [0,n) [ 0 , n ) V V V R ( e ) R(e) R ( e ) Q e ± 1 Q_e^{\pm1} Q e ± 1 ( Q p F ) ± 1 (Q_p^F)^{\pm1} ( Q p F ) ± 1 p ∈ R ( e ) p\in R(e) p ∈ R ( e )
δ ( 1 , Q e ) ≤ ∑ p ∈ R ( e ) δ ( 1 , Q p F ) , ∣ R ( e ) ∣ = s ( t + 1 ) ≤ m ( m + 1 ) . \delta(1,Q_e)\le\sum_{p\in R(e)}\delta(1,Q_p^F),\qquad|R(e)|=s(t+1)\le m(m+1). δ ( 1 , Q e ) ≤ p ∈ R ( e ) ∑ δ ( 1 , Q p F ) , ∣ R ( e ) ∣ = s ( t + 1 ) ≤ m ( m + 1 ) . Multiplicity. If a plaquette p p p R 2 ( e ) R_2(e) R 2 ( e ) { σ , τ } \{\sigma,\tau\} { σ , τ } μ \mu μ p p p y + a e σ + c e τ y+ae_\sigma+ce_\tau y + a e σ + c e τ − m ≤ a , c ≤ m − 1 -m\le a,c\le m-1 − m ≤ a , c ≤ m − 1 p p p μ \mu μ ( 2 m ) 2 (2m)^2 ( 2 m ) 2 ( a , c ) (a,c) ( a , c ) y y y e e e p p p 16 m 2 16m^2 16 m 2 R 2 ( e ) R_2(e) R 2 ( e ) p ∈ R 1 ( e ) p\in R_1(e) p ∈ R 1 ( e ) { σ , μ } \{\sigma,\mu\} { σ , μ } x + a e σ x+ae_\sigma x + a e σ − m ≤ a ≤ m − 1 -m\le a\le m-1 − m ≤ a ≤ m − 1 4 m 4m 4 m e e e 16 m 2 + 4 m = 4 n 2 + 2 n 16m^2+4m=4n^2+2n 16 m 2 + 4 m = 4 n 2 + 2 n R ( e ) R(e) R ( e )
The energy. For the chordal metric δ ( 1 , q ) 2 = 2 − Re Tr q \delta(1,q)^2=2-\operatorname{Re}\operatorname{Tr}q δ ( 1 , q ) 2 = 2 − Re Tr q S = ∑ p ∩ V ≠ ∅ δ ( 1 , Q p F ) 2 S=\sum_{p\cap V\ne\varnothing}\delta(1,Q_p^F)^2 S = ∑ p ∩ V = ∅ δ ( 1 , Q p F ) 2 γ V 1 \gamma_V^1 γ V 1 e − b S / 2 / Z e^{-bS/2}/Z e − b S /2 / Z Z = ∫ e − b S / 2 Z=\int e^{-bS/2} Z = ∫ e − b S /2 S U ( 2 ) V \mathrm{SU}(2)^V SU ( 2 ) V t ↦ log ∫ e − t S t\mapsto\log\int e^{-tS} t ↦ log ∫ e − tS t = 0 t=0 t = 0 t = b / 2 t=b/2 t = b /2 − ∫ S d γ V 1 -\int S\,d\gamma_V^1 − ∫ S d γ V 1 2 log Z / b 2\log Z/b 2 log Z / b V V V δ ( 1 , Q p F ) 2 ≤ 3 ∑ e ∈ p ∩ V δ ( 1 , Q e ) 2 \delta(1,Q_p^F)^2\le3\sum_{e\in p\cap V}\delta(1,Q_e)^2 δ ( 1 , Q p F ) 2 ≤ 3 ∑ e ∈ p ∩ V δ ( 1 , Q e ) 2 S ≤ 18 ∑ e ∈ V δ ( 1 , Q e ) 2 S\le18\sum_{e\in V}\delta(1,Q_e)^2 S ≤ 18 ∑ e ∈ V δ ( 1 , Q e ) 2 Z ≥ I ∣ V ∣ Z\ge I^{|V|} Z ≥ I ∣ V ∣ I = ∫ e − 9 b δ ( 1 , q ) 2 d q I=\int e^{-9b\delta(1,q)^2}\,dq I = ∫ e − 9 b δ ( 1 , q ) 2 d q r = ( 9 b ) − 1 / 2 < 1 r=(9b)^{-1/2}<1 r = ( 9 b ) − 1/2 < 1 r 3 / 24 r^3/24 r 3 /24 I ≥ e − 1 ( 9 b ) − 3 / 2 / 24 I\ge e^{-1}(9b)^{-3/2}/24 I ≥ e − 1 ( 9 b ) − 3/2 /24
∫ S d γ V 1 ≤ 2 ∣ V ∣ L ( b ) b . \int S\,d\gamma_V^1\le\frac{2|V|L(b)}{b}. ∫ S d γ V 1 ≤ b 2∣ V ∣ L ( b ) . Conclusion. The Cauchy–Schwarz inequality inside each disc squares the loop bound:
δ ( 1 , Q e ) 2 ≤ ∣ R ( e ) ∣ ∑ p ∈ R ( e ) δ ( 1 , Q p F ) 2 ≤ m ( m + 1 ) ∑ p ∈ R ( e ) δ ( 1 , Q p F ) 2 . \begin{aligned}
\delta(1,Q_e)^2\le|R(e)|\sum_{p\in R(e)}\delta(1,Q_p^F)^2 \\
&\le m(m+1)\sum_{p\in R(e)}\delta(1,Q_p^F)^2.
\end{aligned} δ ( 1 , Q e ) 2 ≤ ∣ R ( e ) ∣ p ∈ R ( e ) ∑ δ ( 1 , Q p F ) 2 ≤ m ( m + 1 ) p ∈ R ( e ) ∑ δ ( 1 , Q p F ) 2 . Summing over e ∈ V e\in V e ∈ V
∑ e ∈ V δ ( 1 , Q e ) 2 ≤ m ( m + 1 ) ( 4 n 2 + 2 n ) S = n 2 ( n + 2 ) ( 2 n + 1 ) 2 S , \sum_{e\in V}\delta(1,Q_e)^2\le m(m+1)(4n^2+2n)S=\frac{n^2(n+2)(2n+1)}{2}S, e ∈ V ∑ δ ( 1 , Q e ) 2 ≤ m ( m + 1 ) ( 4 n 2 + 2 n ) S = 2 n 2 ( n + 2 ) ( 2 n + 1 ) S , and integrating against γ V 1 \gamma_V^1 γ V 1 ε V \varepsilon_V ε V D ( n , α ) D(n,\alpha) D ( n , α ) 2 ( 1 − α ) < n 2 ( n + 2 ) ( 2 n + 1 ) L ( b ) / b 2(1-\alpha)<n^2(n+2)(2n+1)L(b)/b 2 ( 1 − α ) < n 2 ( n + 2 ) ( 2 n + 1 ) L ( b ) / b ( n + 2 ) ( 2 n + 1 ) ≤ 5 n 2 (n+2)(2n+1)\le5n^2 ( n + 2 ) ( 2 n + 1 ) ≤ 5 n 2 n ≥ 2 n\ge2 n ≥ 2 □ \square □
The loops and their multiplicity use nothing of S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) δ \delta δ S U ( N ) \mathrm{SU}(N) SU ( N ) n n n
Proposition 76. Let G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) N ≥ 2 N\ge2 N ≥ 2 exp ( b N ∑ p Re Tr Q p ) \exp\left(\frac{b}{N}\sum_p\operatorname{Re}\operatorname{Tr}Q_p\right) exp ( N b ∑ p Re Tr Q p ) b = β W ≥ N / 9 b=\beta_W\ge N/9 b = β W ≥ N /9 δ ( p , q ) = ∥ p − q ∥ H S \delta(p,q)=\lVert p-q\rVert_{\mathrm{HS}} δ ( p , q ) = ∥ p − q ∥ HS N ≥ 2 N\ge2 N ≥ 2
L N ( b ) = 1 + N 2 − 1 2 log ( 81 π 2 b ) + N ( N − 1 ) log 12 11 . L_N(b)=1+\frac{N^2-1}{2}\log(81\pi^2b)+N(N-1)\log\frac{12}{11}. L N ( b ) = 1 + 2 N 2 − 1 log ( 81 π 2 b ) + N ( N − 1 ) log 11 12 . Then
1 ∣ V ∣ ∑ e ∈ V ∫ δ ( 1 , Q e ) 2 d γ V 1 ≤ N n 2 ( n + 2 ) ( 2 n + 1 ) L N ( b ) b . \frac{1}{|V|}\sum_{e\in V}\int\delta(1,Q_e)^2\,d\gamma_V^1
\le\frac{Nn^2(n+2)(2n+1)L_N(b)}{b}. ∣ V ∣ 1 e ∈ V ∑ ∫ δ ( 1 , Q e ) 2 d γ V 1 ≤ b N n 2 ( n + 2 ) ( 2 n + 1 ) L N ( b ) . Consequently, if D ( n , α ) D(n,\alpha) D ( n , α ) b b b
n 4 ≥ 2 ( 1 − α ) b 5 L N ( b ) . n^4\ge\frac{2(1-\alpha)b}{5L_N(b)}. n 4 ≥ 5 L N ( b ) 2 ( 1 − α ) b . Proof. Loops. The discs R ( e ) R(e) R ( e ) 4 n 2 + 2 n 4n^2+2n 4 n 2 + 2 n S = ∑ p ∩ V ≠ ∅ δ ( 1 , Q p F ) 2 S=\sum_{p\cap V\ne\varnothing}\delta(1,Q_p^F)^2 S = ∑ p ∩ V = ∅ δ ( 1 , Q p F ) 2 ∑ e ∈ V δ ( 1 , Q e ) 2 ≤ 1 2 n 2 ( n + 2 ) ( 2 n + 1 ) S \sum_{e\in V}\delta(1,Q_e)^2\le\frac12n^2(n+2)(2n+1)S ∑ e ∈ V δ ( 1 , Q e ) 2 ≤ 2 1 n 2 ( n + 2 ) ( 2 n + 1 ) S
The energy. Since δ ( 1 , q ) 2 = 2 N − 2 Re Tr q \delta(1,q)^2=2N-2\operatorname{Re}\operatorname{Tr}q δ ( 1 , q ) 2 = 2 N − 2 Re Tr q γ V 1 \gamma_V^1 γ V 1 e − b S / ( 2 N ) / Z e^{-bS/(2N)}/Z e − b S / ( 2 N ) / Z t ↦ log ∫ e − t S t\mapsto\log\int e^{-tS} t ↦ log ∫ e − tS ∫ S d γ V 1 ≤ 2 N log ( 1 / Z ) / b \int S\,d\gamma_V^1\le2N\log(1/Z)/b ∫ S d γ V 1 ≤ 2 N log ( 1/ Z ) / b S ≤ 18 ∑ e ∈ V δ ( 1 , Q e ) 2 S\le18\sum_{e\in V}\delta(1,Q_e)^2 S ≤ 18 ∑ e ∈ V δ ( 1 , Q e ) 2 Z ≥ I ∣ V ∣ Z\ge I^{|V|} Z ≥ I ∣ V ∣ I = ∫ e − 9 b δ ( 1 , q ) 2 / N d q I=\int e^{-9b\delta(1,q)^2/N}\,dq I = ∫ e − 9 b δ ( 1 , q ) 2 / N d q
Small balls. Give s u ( N ) \mathfrak{su}(N) su ( N ) Tr ( X ∗ Y ) \operatorname{Tr}(X^*Y) Tr ( X ∗ Y ) S U ( N ) \mathrm{SU}(N) SU ( N ) v N v_N v N j ( X ) d X j(X)dX j ( X ) d X d X dX d X
j ( X ) = ∣ det 1 − e − ad X ad X ∣ = ∏ k < l ( sin ( φ k l / 2 ) φ k l / 2 ) 2 . j(X)=\left|\det\frac{1-e^{-\operatorname{ad}X}}{\operatorname{ad}X}\right|=\prod_{k<l}\left(\frac{\sin(\varphi_{kl}/2)}{\varphi_{kl}/2}\right)^2. j ( X ) = det ad X 1 − e − ad X = k < l ∏ ( φ k l /2 sin ( φ k l /2 ) ) 2 . if X X X i θ 1 , … , i θ N i\theta_1,\ldots,i\theta_N i θ 1 , … , i θ N φ k l = θ k − θ l \varphi_{kl}=\theta_k-\theta_l φ k l = θ k − θ l [42 Ch. II, Thm. 1.7] ; in particular j ≤ 1 j\leq1 j ≤ 1 q ∈ S U ( N ) q\in\mathrm{SU}(N) q ∈ SU ( N ) e i ψ k e^{i\psi_k} e i ψ k − π ≤ ψ k < π -\pi\leq\psi_k<\pi − π ≤ ψ k < π ∑ k ψ k = 2 π k 0 \sum_k\psi_k=2\pi k_0 ∑ k ψ k = 2 π k 0 ∣ k 0 ∣ ≤ N / 2 |k_0|\leq N/2 ∣ k 0 ∣ ≤ N /2 ∣ k 0 ∣ |k_0| ∣ k 0 ∣ ψ k \psi_k ψ k ∓ 2 π \mp2\pi ∓ 2 π θ k \theta_k θ k ∣ θ k ∣ ≤ 3 π |\theta_k|\leq3\pi ∣ θ k ∣ ≤ 3 π q = e X q=e^X q = e X X ∈ s u ( N ) X\in\mathfrak{su}(N) X ∈ su ( N ) ∥ X ∥ H S ≤ 3 π N \|X\|_{\mathrm{HS}}\leq3\pi\sqrt{N} ∥ X ∥ HS ≤ 3 π N v N ≤ ω N 2 − 1 ( 3 π N ) N 2 − 1 v_N\leq\omega_{N^2-1}(3\pi\sqrt{N})^{N^2-1} v N ≤ ω N 2 − 1 ( 3 π N ) N 2 − 1 ω k \omega_k ω k R k \mathbb{R}^k R k ∥ X ∥ H S ≤ r ≤ 1 \|X\|_{\mathrm{HS}}\leq r\leq1 ∥ X ∥ HS ≤ r ≤ 1 ∣ θ k ∣ ≤ 1 < π |\theta_k|\leq1<\pi ∣ θ k ∣ ≤ 1 < π exp \exp exp δ ( 1 , e X ) 2 = ∑ k ∣ e i θ k − 1 ∣ 2 ≤ ∥ X ∥ H S 2 \delta(1,e^X)^2=\sum_k|e^{i\theta_k}-1|^2\leq\|X\|_{\mathrm{HS}}^2 δ ( 1 , e X ) 2 = ∑ k ∣ e i θ k − 1 ∣ 2 ≤ ∥ X ∥ HS 2 ∣ φ k l ∣ / 2 ≤ ( θ k 2 + θ l 2 ) 1 / 2 / 2 ≤ 2 − 1 / 2 |\varphi_{kl}|/2\leq(\theta_k^2+\theta_l^2)^{1/2}/\sqrt{2}\leq2^{-1/2} ∣ φ k l ∣/2 ≤ ( θ k 2 + θ l 2 ) 1/2 / 2 ≤ 2 − 1/2 j ( X ) j(X) j ( X ) ( 1 − 1 12 ) 2 (1-\frac{1}{12})^2 ( 1 − 12 1 ) 2 sin x / x ≥ 1 − x 2 / 6 \sin x/x\geq1-x^2/6 sin x / x ≥ 1 − x 2 /6 v N v_N v N { q : δ ( 1 , q ) ≤ r } \{q:\delta(1,q)\leq r\} { q : δ ( 1 , q ) ≤ r }
( 11 12 ) N ( N − 1 ) ω N 2 − 1 r N 2 − 1 v N ≥ ( 11 12 ) N ( N − 1 ) ( r 3 π N ) N 2 − 1 . \left(\frac{11}{12}\right)^{N(N-1)}\frac{\omega_{N^2-1}r^{N^2-1}}{v_N}
\geq\left(\frac{11}{12}\right)^{N(N-1)}
\left(\frac{r}{3\pi\sqrt{N}}\right)^{N^2-1}. ( 12 11 ) N ( N − 1 ) v N ω N 2 − 1 r N 2 − 1 ≥ ( 12 11 ) N ( N − 1 ) ( 3 π N r ) N 2 − 1 . With r = ( N / 9 b ) 1 / 2 ≤ 1 r=(N/9b)^{1/2}\leq1 r = ( N /9 b ) 1/2 ≤ 1 I ≥ e − 1 I\geq e^{-1} I ≥ e − 1 3 π N / r = 9 π b 3\pi\sqrt{N}/r=9\pi\sqrt{b} 3 π N / r = 9 π b log ( 1 / I ) ≤ L N ( b ) \log(1/I)\leq L_N(b) log ( 1/ I ) ≤ L N ( b ) ∫ δ d γ V W ≤ 2 N ∣ V ∣ L N ( b ) / b \int\delta\,d\gamma_V^W\leq2N|V|L_N(b)/b ∫ δ d γ V W ≤ 2 N ∣ V ∣ L N ( b ) / b
Conclusion. Proposition 74 with the defining representation and κ = 1 \kappa=1 κ = 1 δ ( 1 , q ) 2 \delta(1,q)^2 δ ( 1 , q ) 2 2 N 2N 2 N 2 N ( 1 − α ) ≤ N n 2 ( n + 2 ) ( 2 n + 1 ) L N ( b ) / b 2N(1-\alpha)\leq Nn^2(n+2)(2n+1)L_N(b)/b 2 N ( 1 − α ) ≤ N n 2 ( n + 2 ) ( 2 n + 1 ) L N ( b ) / b D ( n , α ) D(n,\alpha) D ( n , α ) ( n + 2 ) ( 2 n + 1 ) ≤ 5 n 2 (n+2)(2n+1)\leq5n^2 ( n + 2 ) ( 2 n + 1 ) ≤ 5 n 2
For every N N N ( β W / log β W ) 1 / 4 (\beta_W/\log\beta_W)^{1/4} ( β W / log β W ) 1/4 N N N L N L_N L N N 2 2 log β W \frac{N^2}{2}\log\beta_W 2 N 2 log β W
The cube condition is a statement about strong coupling at its own scale, and this persists under the perturbations of Proposition 70 when they respect the gauge. Call ( w X ) (w_X) ( w X ) gauge invariant if w X ( Q h ) = w X ( Q ) w_X(Q^h)=w_X(Q) w X ( Q h ) = w X ( Q ) h h h F F F Q ( x , μ ) h = h ( x ) Q ( x , μ ) h ( x + e μ ) − 1 Q^h_{(x,\mu)}=h(x)Q_{(x,\mu)}h(x+e_\mu)^{-1} Q ( x , μ ) h = h ( x ) Q ( x , μ ) h ( x + e μ ) − 1 F F F G A G^A G A ε V W \varepsilon_V^W ε V W ε V \varepsilon_V ε V γ V W ( ⋅ ∣ 1 ) \gamma_V^W(\mathord\cdot\mid1) γ V W ( ⋅ ∣ 1 ) γ V L \gamma_V^L γ V L
Proposition 77. Let ( w X ) (w_X) ( w X ) n = 2 m n=2m n = 2 m D W ( n , α ) D^W(n,\alpha) D W ( n , α )
(i) ε V W ≥ ( 1 − α ) D / 3 \varepsilon_V^W\geq(1-\alpha)D/3 ε V W ≥ ( 1 − α ) D /3
(ii) If η ≥ 0 \eta\geq0 η ≥ 0 0 ≤ κ ≤ 1 0\leq\kappa\leq1 0 ≤ κ ≤ 1 δ ( 1 , Q p F ) ≤ η \delta(1,Q_p^F)\leq\eta δ ( 1 , Q p F ) ≤ η p p p V V V 1 − κ 1-\kappa 1 − κ γ V W ( ⋅ ∣ 1 ) \gamma_V^W(\mathord\cdot\mid1) γ V W ( ⋅ ∣ 1 )
m ( m + 1 ) η + κ D ≥ ( 1 − α ) D 3 . m(m+1)\eta+\kappa D\geq\frac{(1-\alpha)D}{3}. m ( m + 1 ) η + κ D ≥ 3 ( 1 − α ) D . Proof. (i) The proof of Theorem 71 uses the action only through the identity γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V L \gamma_V(\mathord\cdot\mid Q^g)=\Phi_*\gamma_V^L γ V ( ⋅ ∣ Q g ) = Φ ∗ γ V L C V , j C_{V,j} C V , j g g g F F F g g g w X w_X w X γ V W \gamma_V^W γ V W V V V γ V W \gamma_V^W γ V W X X X C V , j C_{V,j} C V , j K C W K_C^W K C W C V , j C_{V,j} C V , j K C K_C K C
(ii) Under γ V W ( ⋅ ∣ 1 ) \gamma_V^W(\mathord\cdot\mid1) γ V W ( ⋅ ∣ 1 ) V V V V V V δ \delta δ δ ( 1 , Q e ) ≤ ∑ p ∈ R ( e ) δ ( 1 , Q p F ) ≤ ( m + 1 ) η \delta(1,Q_e)\leq\sum_{p\in R(e)}\delta(1,Q_p^F)\leq(m+1)\eta δ ( 1 , Q e ) ≤ ∑ p ∈ R ( e ) δ ( 1 , Q p F ) ≤ ( m + 1 ) η e ∈ V e\in V e ∈ V δ ( 1 , Q e ) ≤ D \delta(1,Q_e)\leq D δ ( 1 , Q e ) ≤ D e ∈ V e\in V e ∈ V ε V W ≤ m ( m + 1 ) η + κ D \varepsilon_V^W\leq m(m+1)\eta+\kappa D ε V W ≤ m ( m + 1 ) η + κ D
In the small-field region of a gauge-covariant block renormalisation the blocked plaquette variables lie within a distance η \eta η κ \kappa κ κ < ( 1 − α ) / 3 \kappa<(1-\alpha)/3 κ < ( 1 − α ) /3 m ( m + 1 ) ≥ x m(m+1)\geq x m ( m + 1 ) ≥ x x = ( ( 1 − α ) / 3 − κ D ) / η x=((1-\alpha)/3-\kappa D)/\eta x = (( 1 − α ) /3 − κ D ) / η n − 1 / 2 n^{-1/2} n − 1/2 n N = m 4 ≥ x 2 / 4 n_N=m^4\geq x^2/4 n N = m 4 ≥ x 2 /4 α ′ ≥ α + 96 s Ω \alpha'\geq\alpha+96s\Omega α ′ ≥ α + 96 s Ω α ′ < 1 \alpha'<1 α ′ < 1
ε < 1 − α 4608 s M D N ≤ ( 1 − α ) η 2 1152 s M D 3 ( ( 1 − α ) / 3 − κ ) 2 , \begin{aligned}
\varepsilon<\frac{1-\alpha}{4608sMDN} \\
&\leq\frac{(1-\alpha)\eta^2}{1152sMD^3((1-\alpha)/3-\kappa)^2},
\end{aligned} ε < 4608 s M D N 1 − α ≤ 1152 s M D 3 (( 1 − α ) /3 − κ ) 2 ( 1 − α ) η 2 , which is of order η 2 \eta^2 η 2
The next statement shows that the cube condition separates a theory with a massless photon from the expected behaviour of S U ( N ) \mathrm{SU}(N) SU ( N ) G = U ( 1 ) G=\mathrm{U}(1) G = U ( 1 ) D = 2 D=2 D = 2
φ β ( θ ) = ∑ k ∈ Z e − β ( θ + 2 π k ) 2 / 2 , \varphi_\beta(\theta)=\sum_{k\in\mathbb{Z}}e^{-\beta(\theta+2\pi k)^2/2}, φ β ( θ ) = k ∈ Z ∑ e − β ( θ + 2 π k ) 2 /2 , so that the measure of a box Λ \Lambda Λ ∏ p ∈ P ( Λ ) φ β ( θ p ) \prod_{p\in P(\Lambda)}\varphi_\beta(\theta_p) ∏ p ∈ P ( Λ ) φ β ( θ p ) e i θ p e^{i\theta_p} e i θ p F F \mathcal{F}^F F F C V , j C_{V,j} C V , j K C K_C K C D ( n , α ) D(n,\alpha) D ( n , α ) φ β \varphi_\beta φ β
Proposition 78. There is β 0 \beta_0 β 0 β > β 0 \beta>\beta_0 β > β 0 D ( n , α ) D(n,\alpha) D ( n , α ) n ≥ 2 n\geq2 n ≥ 2 α < 1 \alpha<1 α < 1 β K → ∞ \beta_K\to\infty β K → ∞ n k n_k n k Theorem 45 fails for this theory.
Proof. Lemma 10 , Proposition 11(i)--(iii) and (v), Lemma 112, Corollary 113 and Lemma 115 use the Wilson weight only as a positive continuous class function of the plaquette variable, with one exception: the Doeblin bound in Lemma 115(a), which for the Villain weight reads that each single-link density is at least ( min φ β / max φ β ) 6 > 0 (\min\varphi_\beta/\max\varphi_\beta)^6>0 ( min φ β / max φ β ) 6 > 0 D ( n , α ) D(n,\alpha) D ( n , α ) Theorem 43 , the limit ω ( O ) = lim W Λ ( O ) \omega(O)=\lim W_{\Lambda}(O) ω ( O ) = lim W Λ ( O ) Z 4 \mathbb{Z}^4 Z 4 O , O ′ O,O' O , O ′ O O O
∣ Cov ω ( O , O ′ ) ∣ ≤ ∥ O ′ ∥ ∞ L O D ( 1 + c n e λ ρ n ) 1 − θ ∣ S O ′ ∣ e − λ dist ( O , O ′ ) \left|\operatorname{Cov}_{\omega}(O,O')\right|\leq\|O'\|_{\infty}L_O\frac{D(1+c_ne^{\lambda\rho_n})}{1-\theta}|S_{O'}|e^{-\lambda\operatorname{dist}(O,O')} ∣ Cov ω ( O , O ′ ) ∣ ≤ ∥ O ′ ∥ ∞ L O 1 − θ D ( 1 + c n e λ ρ n ) ∣ S O ′ ∣ e − λ dist ( O , O ′ ) for every λ > 0 \lambda>0 λ > 0 θ = α e λ ρ n < 1 \theta=\alpha e^{\lambda\rho_n}<1 θ = α e λ ρ n < 1 L O L_O L O O O O S O ′ S_{O'} S O ′ O ′ O' O ′
Put Ψ p = φ β ′ ( θ p ) / φ β ( θ p ) \Psi_p=\varphi'_{\beta}(\theta_p)/\varphi_{\beta}(\theta_p) Ψ p = φ β ′ ( θ p ) / φ β ( θ p ) − i -i − i [36 Sect. 2] for the Villain action. It is real, odd and smooth in θ p \theta_p θ p p p p ω ( Ψ p ) = 0 \omega(\Psi_p)=0 ω ( Ψ p ) = 0 θ ↦ − θ \theta\mapsto-\theta θ ↦ − θ ∑ p ′ ∣ ω ( Ψ p Ψ p ′ ) ∣ < ∞ \sum_{p'}|\omega(\Psi_p\Psi_{p'})|<\infty ∑ p ′ ∣ ω ( Ψ p Ψ p ′ ) ∣ < ∞ p ′ p' p ′ p p p [36 ] is the limit of the same free-boundary measures over boxes, which is ω \omega ω β \beta β β 0 \beta_0 β 0 [41 ] . This contradiction proves the first statement, and the second follows because β k > β 0 \beta_k>\beta_0 β k > β 0 k k k
At large β \beta β SU ( N ) \operatorname{SU}(N) SU ( N ) G G G β \beta β [36 Sect. 2] , so the proposition is stated for the Villain action only.
Remark 79. The proof loses the area of a disc, of order n 2 n^2 n 2 ( Z / L Z ) 4 (\mathbb{Z}/L\mathbb{Z})^4 ( Z / L Z ) 4 L L L 0 ≤ x j < L 0\leq x_j<L 0 ≤ x j < L ϕ = 4 π / L 2 \phi=4\pi/L^2 ϕ = 4 π / L 2 q ( θ ) = cos θ + i sin θ q(\theta)=\cos\theta+i\sin\theta q ( θ ) = cos θ + i sin θ U ( x , 2 ) = q ( ϕ x 1 ) U_{(x,2)}=q(\phi x_1) U ( x , 2 ) = q ( ϕ x 1 ) U ( x , 1 ) = q ( − ϕ L x 2 ) U_{(x,1)}=q(-\phi Lx_2) U ( x , 1 ) = q ( − ϕ L x 2 ) x 1 = L − 1 x_1=L-1 x 1 = L − 1 U = 1 U=1 U = 1 { 1 , 2 } \{1,2\} { 1 , 2 } q ( ϕ ) q(\phi) q ( ϕ ) x 1 = L − 1 x_1=L-1 x 1 = L − 1 x 1 = x 2 = L − 1 x_1=x_2=L-1 x 1 = x 2 = L − 1 ϕ − 4 π \phi-4\pi ϕ − 4 π ϕ \phi ϕ [ 0 , L / 2 ] 2 [0,L/2]^2 [ 0 , L /2 ] 2 { 1 , 2 } \{1,2\} { 1 , 2 } q ( ϕ L 2 / 4 ) = − 1 q(\phi L^2/4)=-1 q ( ϕ L 2 /4 ) = − 1 D = 2 D=2 D = 2 ϕ L 2 / 4 = π \phi L^2/4=\pi ϕ L 2 /4 = π [55 ] does not change it. Plaquette variables within O ( n − 2 ) O(n^{-2}) O ( n − 2 ) n n n o ( n − 2 ) o(n^{-2}) o ( n − 2 )
Up to the factor L ( b ) 1 / 2 L(b)^{1/2} L ( b ) 1/2 b − 1 / 2 b^{-1/2} b − 1/2 n n n
Proposition 80. Let G = SU ( 2 ) G=\operatorname{SU}(2) G = SU ( 2 ) b = β W > 0 b=\beta_W>0 b = β W > 0 e e e Q e Q_e Q e
∫ δ ( 1 , Q e ) d μ e ≥ 1 9 max ( 6 b , 1 ) 1 / 2 . \int\delta(1,Q_e)\,d\mu_e\geq\frac{1}{9\max(6b,1)^{1/2}}. ∫ δ ( 1 , Q e ) d μ e ≥ 9 max ( 6 b , 1 ) 1/2 1 . In particular ϵ V ( β ) ≥ 1 9 max ( 6 b , 1 ) − 1 / 2 \epsilon_V(\beta)\geq\frac{1}{9}\max(6b,1)^{-1/2} ϵ V ( β ) ≥ 9 1 max ( 6 b , 1 ) − 1/2 n n n n n n ϵ V ( β ) \epsilon_V(\beta) ϵ V ( β ) b − 1 / 2 b^{-1/2} b − 1/2 ( L ( b ) / b ) 1 / 2 (L(b)/b)^{1/2} ( L ( b ) / b ) 1/2
Proof. Write ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ Re Tr q = 2 ⟨ q , 1 ⟩ \operatorname{Re}\operatorname{Tr}q=2\langle q,1\rangle Re Tr q = 2 ⟨ q , 1 ⟩ e e e exp ( b ⟨ Q e , W p ⟩ ) \exp(b\langle Q_e,W_p\rangle) exp ( b ⟨ Q e , W p ⟩) ∣ W p ∣ = 1 |W_p|=1 ∣ W p ∣ = 1 μ e \mu_e μ e e ⟨ v , q ⟩ / Z ( ∣ v ∣ ) e^{\langle v,q\rangle}/Z(|v|) e ⟨ v , q ⟩ / Z ( ∣ v ∣ ) v v v a = ∣ v ∣ ≤ 6 b a=|v|\leq6b a = ∣ v ∣ ≤ 6 b a ′ = max ( a , 1 ) a'=\max(a,1) a ′ = max ( a , 1 ) η = a ′ − 1 / 2 ≤ 1 \eta=a'^{-1/2}\leq1 η = a ′ − 1/2 ≤ 1 η \eta η v / ∣ v ∣ v/|v| v /∣ v ∣ a ( 1 − η 2 / 2 ) ≥ a − 1 2 a(1-\eta^2/2)\geq a-\frac12 a ( 1 − η 2 /2 ) ≥ a − 2 1 η 3 / 24 \eta^3/24 η 3 /24 Z ( a ) ≥ e a − 1 / 2 η 3 / 24 Z(a)\geq e^{a-1/2}\eta^3/24 Z ( a ) ≥ e a − 1/2 η 3 /24 24 e 1 / 2 a ′ 3 / 2 24e^{1/2}a'^{3/2} 24 e 1/2 a ′ 3/2 a < 1 a<1 a < 1 Z ≥ 1 Z\geq1 Z ≥ 1 r r r θ \theta θ 2 sin ( θ / 2 ) = r 2\sin(\theta/2)=r 2 sin ( θ /2 ) = r θ ≤ π r / 2 \theta\leq\pi r/2 θ ≤ π r /2 ( θ − sin θ cos θ ) / π ≤ 2 θ 3 / ( 3 π ) ≤ π 2 r 3 / 12 (\theta-\sin\theta\cos\theta)/\pi\leq2\theta^3/(3\pi)\leq\pi^2r^3/12 ( θ − sin θ cos θ ) / π ≤ 2 θ 3 / ( 3 π ) ≤ π 2 r 3 /12 r = 1 2 a ′ − 1 / 2 r=\frac12a'^{-1/2} r = 2 1 a ′ − 1/2
μ e ( δ ( 1 , Q e ) < r ) ≤ 24 e 1 / 2 a ′ 3 / 2 π 2 r 3 12 = 2 e 1 / 2 π 2 ( 2 9 ) 3 < 1 2 , \mu_e\bigl(\delta(1,Q_e)<r\bigr)\leq24e^{1/2}a'^{3/2}\frac{\pi^2r^3}{12}=2e^{1/2}\pi^2\left(\frac{2}{9}\right)^3<\frac12, μ e ( δ ( 1 , Q e ) < r ) ≤ 24 e 1/2 a ′ 3/2 12 π 2 r 3 = 2 e 1/2 π 2 ( 9 2 ) 3 < 2 1 , so the mean of δ ( 1 , Q e ) \delta(1,Q_e) δ ( 1 , Q e ) r / 2 = 1 4 a ′ − 1 / 2 r/2=\frac14a'^{-1/2} r /2 = 4 1 a ′ − 1/2 e ∈ V e\in V e ∈ V ϵ V \epsilon_V ϵ V
Along the continuum tuning Theorem 71 says the following. If (D) holds, then in a cube of fixed physical size ℓ \ell ℓ ( 1 − α ) D / 3 (1-\alpha)D/3 ( 1 − α ) D /3 1 − α 1-\alpha 1 − α SU ( 2 ) \operatorname{SU}(2) SU ( 2 ) n = ℓ / a n=\ell/a n = ℓ / a 1 / a 1/a 1/ a β ( a ) \beta(a) β ( a ) log ( 1 / a ) \log(1/a) log ( 1/ a ) n n n log ( 1 / a ) \log(1/a) log ( 1/ a )
The last statement concerns the following strategy: control the conditional law of one line of links, given everything else, uniformly in the configuration around it, and assemble the cube from lines. Fix a site x x x s = σ ( x ) s=\sigma(x) s = σ ( x ) μ ≠ s \mu\ne s μ = s c k = ( x + k e s , μ ) c_k=(x+ke_s,\mu) c k = ( x + k e s , μ ) k ∈ Z k\in\mathbb{Z} k ∈ Z c k c_k c k σ \sigma σ x x x
Proposition 81. Let G = SU ( 2 ) G=\operatorname{SU}(2) G = SU ( 2 ) b = β W ≥ 1 b=\beta_W\geq1 b = β W ≥ 1 { c k } \{c_k\} { c k } b b b ( Q c k ) k (Q_{c_k})_k ( Q c k ) k
positive self-adjoint operator K b K_b K b L 2 ( S 3 ) L^2(S^3) L 2 ( S 3 ) Λ b \Lambda_b Λ b
0 < 1 − Λ b , 1 Λ b ≤ e 20 24 10 b 15 e − b / 1280 , 0 < 1-\frac{\Lambda_{b,1}}{\Lambda_b} \le e^{20}24^{10}b^{15}e^{-b/1280}, 0 < 1 − Λ b Λ b , 1 ≤ e 20 2 4 10 b 15 e − b /1280 , where Λ b , 1 \Lambda_{b,1} Λ b , 1
Proof. The interactions. By Lemma 9(iii), c k c_k c k { μ , s } \{\mu,s\} { μ , s } c k − 1 c_{k-1} c k − 1 c k + 1 c_{k+1} c k + 1 ( x + k e s , s ) (x+ke_s,s) ( x + k e s , s ) ( x + ( k − 1 ) e s , s ) (x+(k-1)e_s,s) ( x + ( k − 1 ) e s , s ) F F F s s s { μ , t } \{\mu,t\} { μ , t } { μ , ρ } \{\mu,\rho\} { μ , ρ } c k c_k c k F F F c k c_k c k s s s x s + k x_s+k x s + k k k k c k c_k c k s s s x + k e s x+ke_s x + k e s { μ , s } \{\mu,s\} { μ , s } b ⟨ q k , q k + 1 ⟩ b\langle q_k,q_{k+1}\rangle b ⟨ q k , q k + 1 ⟩ q k = Q c k q_k=Q_{c_k} q k = Q c k c k c_k c k X X X q k X ± 1 q_kX^{\pm1} q k X ± 1 q k − 1 X ± 1 q_k^{-1}X^{\pm1} q k − 1 X ± 1 Re ( q u ˉ ) = Re ( q ˉ u ) = ⟨ q , u ⟩ \operatorname{Re}(q\bar u)=\operatorname{Re}(\bar q u)=\langle q,u\rangle Re ( q u ˉ ) = Re ( q ˉ u ) = ⟨ q , u ⟩ X X X b ⟨ q k , u ⟩ b\langle q_k,u\rangle b ⟨ q k , u ⟩ u u u t ± ( 1 − ∣ t ∣ 2 ) 1 / 2 ν t\pm(1-|t|^2)^{1/2}\nu t ± ( 1 − ∣ t ∣ 2 ) 1/2 ν t = v / 4 t=v/4 t = v /4 ν ⊥ t \nu\perp t ν ⊥ t exp ( b ∑ k ⟨ q k , q k + 1 ⟩ + ∑ k ⟨ h k , q k ⟩ ) \exp\left(b\sum_k\langle q_k,q_{k+1}\rangle+\sum_k\langle h_k,q_k\rangle\right) exp ( b ∑ k ⟨ q k , q k + 1 ⟩ + ∑ k ⟨ h k , q k ⟩ ) h k h_k h k ∣ h k ∣ ≤ 4 b |h_k|\le4b ∣ h k ∣ ≤ 4 b
The fields. Let a = 1 2 a=\frac12 a = 2 1 c = 3 2 c=\frac{\sqrt3}{2} c = 2 3 e 1 , … , e 4 e_1,\ldots,e_4 e 1 , … , e 4 R 4 \mathbb{R}^4 R 4 r k r_k r k ( r 0 , … , r 9 ) = a ( e 1 , − e 1 , e 2 , − e 2 , e 3 , − e 3 , e 3 , − e 2 , e 2 , − e 1 ) (r_0,\ldots,r_9)=a(e_1,-e_1,e_2,-e_2,e_3,-e_3,e_3,-e_2,e_2,-e_1) ( r 0 , … , r 9 ) = a ( e 1 , − e 1 , e 2 , − e 2 , e 3 , − e 3 , e 3 , − e 2 , e 2 , − e 1 ) h k = b ( 2 r k − r k − 1 − r k + 1 ) h_k=b(2r_k-r_{k-1}-r_{k+1}) h k = b ( 2 r k − r k − 1 − r k + 1 ) ∣ h k ∣ ≤ 4 a b = 2 b |h_k|\le4ab=2b ∣ h k ∣ ≤ 4 ab = 2 b b b b h k / b h_k/b h k / b v k = q k − r k v_k=q_k-r_k v k = q k − r k ∣ q k ∣ = 1 |q_k|=1 ∣ q k ∣ = 1 ∣ r k ∣ = a |r_k|=a ∣ r k ∣ = a 0 ≤ k ≤ 9 0\le k\le9 0 ≤ k ≤ 9
1 2 ∑ k ∣ v k + 1 − v k ∣ 2 = ∑ k ( 1 − ⟨ q k , q k + 1 ⟩ ) − 1 b ∑ k ⟨ h k , q k ⟩ + 1 2 ∑ k ∣ r k + 1 − r k ∣ 2 + B ( q 0 ) − B ( q 10 ) , \frac12\sum_k|v_{k+1}-v_k|^2=\sum_k(1-\langle q_k,q_{k+1}\rangle)-\frac1b\sum_k\langle h_k,q_k\rangle+\frac12\sum_k|r_{k+1}-r_k|^2+B(q_0)-B(q_{10}), 2 1 k ∑ ∣ v k + 1 − v k ∣ 2 = k ∑ ( 1 − ⟨ q k , q k + 1 ⟩) − b 1 k ∑ ⟨ h k , q k ⟩ + 2 1 k ∑ ∣ r k + 1 − r k ∣ 2 + B ( q 0 ) − B ( q 10 ) , with B ( q ) = ⟨ q , r 0 − r 9 ⟩ B(q)=\langle q,r_0-r_9\rangle B ( q ) = ⟨ q , r 0 − r 9 ⟩
K b ( x , y ) = ∫ ( S 3 ) 9 exp ( − b 2 ∑ k = 0 9 ∣ v k + 1 − v k ∣ 2 ) d q 1 ⋯ d q 9 , K_b(x,y)=\int_{(S^3)^9}\exp\left(-\frac b2\sum_{k=0}^{9}|v_{k+1}-v_k|^2\right)dq_1\cdots dq_9, K b ( x , y ) = ∫ ( S 3 ) 9 exp ( − 2 b k = 0 ∑ 9 ∣ v k + 1 − v k ∣ 2 ) d q 1 ⋯ d q 9 , with q 0 = x q_0=x q 0 = x q 10 = y q_{10}=y q 10 = y r 0 , … , r 10 r_0,\ldots,r_{10} r 0 , … , r 10 K b = B b B b ∗ K_b=B_bB_b^* K b = B b B b ∗ B b B_b B b K b K_b K b Λ b \Lambda_b Λ b ψ b > 0 \psi_b>0 ψ b > 0 x 4 ↦ − x 4 x_4\mapsto-x_4 x 4 ↦ − x 4 K b K_b K b
A barrier. Let E = 1 2 ∑ k = 0 9 ∣ v k + 1 − v k ∣ 2 E=\frac12\sum_{k=0}^{9}|v_{k+1}-v_k|^2 E = 2 1 ∑ k = 0 9 ∣ v k + 1 − v k ∣ 2 ( q 0 ) 4 ≥ 0 ≥ ( q 10 ) 4 (q_0)_4\ge0\ge(q_{10})_4 ( q 0 ) 4 ≥ 0 ≥ ( q 10 ) 4 E ≤ 1 1280 E\le\frac1{1280} E ≤ 1280 1 Δ = ( 20 E ) 1 / 2 ≤ 1 8 \Delta=(20E)^{1/2}\le\frac18 Δ = ( 20 E ) 1/2 ≤ 8 1 w = v 0 w=v_0 w = v 0 ∣ w − v k ∣ ≤ Δ |w-v_k|\le\Delta ∣ w − v k ∣ ≤ Δ k k k 0 ≤ w 4 ≤ Δ 0\le w_4\le\Delta 0 ≤ w 4 ≤ Δ r k r_k r k ∣ v k + r k ∣ ≤ 1 |v_k+r_k|\le1 ∣ v k + r k ∣ ≤ 1 ∣ w + r k ∣ 2 − 1 ≤ Δ ( 2 + Δ ) = : η |w+r_k|^2-1\le\Delta(2+\Delta)=:\eta ∣ w + r k ∣ 2 − 1 ≤ Δ ( 2 + Δ ) =: η 0 ≤ k ≤ 5 0\le k\le5 0 ≤ k ≤ 5 r k = ± a e l r_k=\pm ae_l r k = ± a e l ∣ w l ∣ ≤ η / ( 2 a ) = η |w_l|\le\eta/(2a)=\eta ∣ w l ∣ ≤ η / ( 2 a ) = η l = 1 , 2 , 3 a n d ∣ w ∣ 2 ≥ 3 4 − η l=1,2,3 and |w|^2\ge\frac34-\eta l = 1 , 2 , 3 an d ∣ w ∣ 2 ≥ 4 3 − η 3 4 ≤ 3 η 2 + Δ 2 + η ≤ 3 ( 17 64 ) 2 + 1 64 + 17 64 = 2019 4096 \frac34\le3\eta^2+\Delta^2+\eta\le3\left(\frac{17}{64}\right)^2+\frac1{64}+\frac{17}{64}=\frac{2019}{4096} 4 3 ≤ 3 η 2 + Δ 2 + η ≤ 3 ( 64 17 ) 2 + 64 1 + 64 17 = 4096 2019 K b ( x , y ) ≤ e − b / 1280 K_b(x,y)\le e^{-b/1280} K b ( x , y ) ≤ e − b /1280 x 4 ≥ 0 ≥ y 4 x_4\ge0\ge y_4 x 4 ≥ 0 ≥ y 4
The top eigenvalue. Let ε = b − 1 / 2 \varepsilon=b^{-1/2} ε = b − 1/2 ε ≤ 1 \varepsilon\le1 ε ≤ 1 S 3 S^3 S 3 2 π ∫ 0 ε sin 2 θ d θ ≥ 8 ε 3 3 π 2 > ε 3 4 \frac2\pi\int_0^\varepsilon\sin^2\theta\,d\theta\ge\frac{8\varepsilon^3}{3\pi^2}>\frac{\varepsilon^3}{4} π 2 ∫ 0 ε sin 2 θ d θ ≥ 3 π 2 8 ε 3 > 4 ε 3 q k q_k q k ε \varepsilon ε r k + c e 4 r_k+ce_4 r k + c e 4 ∣ v k − c e 4 ∣ ≤ ε |v_k-ce_4|\le\varepsilon ∣ v k − c e 4 ∣ ≤ ε E ≤ 20 ε 2 E\le20\varepsilon^2 E ≤ 20 ε 2 e − 20 e^{-20} e − 20 K b K_b K b r 0 + c e 4 r_0+ce_4 r 0 + c e 4 Λ b ≥ e − 20 ( ε 3 / 4 ) 10 = e − 20 24 − 10 b − 15 \Lambda_b\ge e^{-20}(\varepsilon^3/4)^{10}=e^{-20}24^{-10}b^{-15} Λ b ≥ e − 20 ( ε 3 /4 ) 10 = e − 20 2 4 − 10 b − 15
The gap. Let ϕ ( x ) = sign ( x 4 ) ψ b ( x ) \phi(x)=\operatorname{sign}(x_4)\psi_b(x) ϕ ( x ) = sign ( x 4 ) ψ b ( x ) ψ b \psi_b ψ b
Λ b − ⟨ ϕ , K b ϕ ⟩ = 4 ∫ x 4 > 0 > y 4 ψ b ( x ) K b ( x , y ) ψ b ( y ) d x d y ≤ e − b / 1280 , \begin{aligned}
\Lambda_b-\langle\phi,K_b\phi\rangle=4\int_{x_4>0>y_4}\psi_b(x)K_b(x,y)\psi_b(y)\,dx\,dy \\
&\le e^{-b/1280},
\end{aligned} Λ b − ⟨ ϕ , K b ϕ ⟩ = 4 ∫ x 4 > 0 > y 4 ψ b ( x ) K b ( x , y ) ψ b ( y ) d x d y ≤ e − b /1280 , because ∫ x 4 > 0 ψ b ≤ 1 2 \int_{x_4>0}\psi_b\le\frac12 ∫ x 4 > 0 ψ b ≤ 2 1 Λ b , 1 ≥ ⟨ ϕ , K b ϕ ⟩ \Lambda_{b,1}\ge\langle\phi,K_b\phi\rangle Λ b , 1 ≥ ⟨ ϕ , K b ϕ ⟩ Λ b \Lambda_b Λ b Λ b \Lambda_b Λ b
The exterior of Proposition 81 is legal at every coupling, so a bound on the conditional law of single lines that is uniform in the exterior cannot be better than e − b / 1280 e^{-b/1280} e − b /1280 b b b C V , J C_{V,J} C V , J
Block averages
Proposition 63 locates the failure of (D) in the quadratic approximation: gauge transformations constant on the lines of F F F [53 ] of the third author, which keeps the cube condition but changes the gauge and conditions the cube on block averages of the field, as in Balaban’s renormalisation group [5 , 6 ] . The natural block averages of the line field leave one class of lines invisible and fail at every size (Proposition 82). In a gauge whose fixed links are trees inside blocks of side two, the averages over whole blocks remove every zero mode, the constrained quadratic form has a gap that does not depend on the volume (Theorem 85), and the constrained cube condition holds at every coupling once the cube is large (Theorem 88). Integrating out one scale bounds the cube coefficients by the constrained ones plus those of the averages (Corollary 91 ), and the averages carry at least the coarse quadratic form at every scale (Proposition 89). For the group-valued links themselves, the straight paths between block roots satisfy a coercivity bound for all fields but do not compose, a continuous covariant map cannot treat the transverse positions alike, and a cut-off of Balaban’s averaging operation is coercive for all fields and agrees with the block averages to first order (Sec. XII B). It composes across all scales on fields of small curvature for every compact group (Sec. XII C), and under the SU(2) Wilson measure the fields that are not small form clusters of vanishing physical size (Sec. XII D).
Block-constrained cubes
In this subsection everything concerns the quadratic approximation: real fields φ \varphi φ ( d φ ) p (d\varphi)_p ( d φ ) p p p p ( d h ) ( x , μ ) = h ( x ) − h ( x + e μ ) (dh)_{(x,\mu)}=h(x)-h(x+e_\mu) ( d h ) ( x , μ ) = h ( x ) − h ( x + e μ ) h h h
Proposition 82 ([53 ] ). In the setting of Proposition 63 , condition the cube in addition on any family of the face averages
V y , μ f a c e ( φ ) = 1 8 ∑ x ( φ ( x , μ ) + φ ( x + e μ , μ ) ) , V^{\mathrm{face}}_{y,\mu}(\varphi)=\frac{1}{8}\sum_x\left(\varphi_{(x,\mu)}+\varphi_{(x+e_\mu,\mu)}\right), V y , μ face ( φ ) = 8 1 x ∑ ( φ ( x , μ ) + φ ( x + e μ , μ ) ) , where x x x 2 y + { 0 , 1 } 4 2y+\{0,1\}^4 2 y + { 0 , 1 } 4 x μ = 2 y μ x_\mu=2y_\mu x μ = 2 y μ ∑ c ∈ C I K c ≥ 12 N n \sum_{c\in C_I}K_c\ge12N_n ∑ c ∈ C I K c ≥ 12 N n
C I = { ( 0111 , μ ) , ( 1111 , μ ) , ( 0111 + e μ , μ ) , ( 1111 + e μ , μ ) : μ = 1 , 2 , 3 } , C_I=\{(0111,\mu),(1111,\mu),(0111+e_\mu,\mu),(1111+e_\mu,\mu):\mu=1,2,3\}, C I = {( 0111 , μ ) , ( 1111 , μ ) , ( 0111 + e μ , μ ) , ( 1111 + e μ , μ ) : μ = 1 , 2 , 3 } , written as residue of the base and direction. Hence K c ≥ N n K_c\ge N_n K c ≥ N n α < 1 \alpha<1 α < 1 n n n b b b
Proof. By tbl:III , σ ( 0111 ) = σ ( 1111 ) = 0 \sigma(0111)=\sigma(1111)=0 σ ( 0111 ) = σ ( 1111 ) = 0 F F F x 1 , x 2 , x 3 x_1,x_2,x_3 x 1 , x 2 , x 3 ℓ \ell ℓ h ℓ h_\ell h ℓ d h ℓ dh_\ell d h ℓ x x x h ℓ ( x ) − h ℓ ( x + 2 e μ ) h_\ell(x)-h_\ell(x+2e_\mu) h ℓ ( x ) − h ℓ ( x + 2 e μ ) μ ≠ 0 \mu\ne0 μ = 0 μ \mu μ ℓ \ell ℓ μ = 0 \mu=0 μ = 0 ℓ \ell ℓ d h ℓ dh_\ell d h ℓ ∑ i ∈ V ∣ ( d h ℓ ) i ∣ ≤ ∑ j ∉ V C V , j ∣ ( d h ℓ ) j ∣ \sum_{i\in V}|(dh_\ell)_i|\le\sum_{j\notin V}C_{V,j}|(dh_\ell)_j| ∑ i ∈ V ∣ ( d h ℓ ) i ∣ ≤ ∑ j ∈ / V C V , j ∣ ( d h ℓ ) j ∣ e = ( x , μ ) e=(x,\mu) e = ( x , μ ) ∣ ( d h ℓ ) e ∣ = 1 |(dh_\ell)_e|=1 ∣ ( d h ℓ ) e ∣ = 1 ℓ \ell ℓ μ ≠ 0 \mu\ne0 μ = 0 μ \mu μ μ = 0 \mu=0 μ = 0 F F F ℓ \ell ℓ tbl:III are the classes of C I C_I C I μ ≠ 0 \mu\ne0 μ = 0 r + e μ r+e_\mu r + e μ r ∈ { 0111 , 1111 } r\in\{0111,1111\} r ∈ { 0111 , 1111 } μ \mu μ σ ( r + e μ ) \sigma(r+e_\mu) σ ( r + e μ ) σ ( 0011 ) = 3 ≠ 1 \sigma(0011)=3\ne1 σ ( 0011 ) = 3 = 1 12 N n ≤ ∑ c ∈ C I K c 12N_n\le\sum_{c\in C_I}K_c 12 N n ≤ ∑ c ∈ C I K c
Definition 83. The blocks are B ( y ) = 2 y + { 0 , 1 } 4 B(y)=2y+\{0,1\}^4 B ( y ) = 2 y + { 0 , 1 } 4 y ∈ Z 4 y\in\mathbb{Z}^4 y ∈ Z 4 o = x m o d 2 o=x\bmod2 o = x mod 2 ( x , μ ) (x,\mu) ( x , μ ) tree link if o μ = 0 o_\mu=0 o μ = 0 o ν = 0 o_\nu=0 o ν = 0 ν > μ \nu>\mu ν > μ active otherwise; fields vanish on tree links. For a coarse link ( y , μ ) (y,\mu) ( y , μ ) block average is
V y , μ ( φ ) = 1 16 ∑ x ∈ B ( y ) ( φ ( x , μ ) + φ ( x + e μ , μ ) ) , V_{y,\mu}(\varphi)=\frac{1}{16}\sum_{x\in B(y)}\left(\varphi_{(x,\mu)}+\varphi_{(x+e_\mu,\mu)}\right), V y , μ ( φ ) = 16 1 x ∈ B ( y ) ∑ ( φ ( x , μ ) + φ ( x + e μ , μ ) ) , the mean of φ \varphi φ μ \mu μ B ( y ) B(y) B ( y )
Lemma 84. (i) The fifteen tree links of a block form a spanning tree of its sixteen sites. Each block carries 49 active links: the 17 active links inside it, with o μ = 0 o_\mu=0 o μ = 0 o μ = 1 o_\mu=1 o μ = 1 B ( y ) B(y) B ( y ) B ( y + e μ ) B(y+e_\mu) B ( y + e μ )
(ii) If d h dh d h h h h V y , μ ( φ + d h ) = V y , μ ( φ ) + h B ( y ) − h B ( y + e μ ) V_{y,\mu}(\varphi+dh)=V_{y,\mu}(\varphi)+h_{B(y)}-h_{B(y+e_\mu)} V y , μ ( φ + d h ) = V y , μ ( φ ) + h B ( y ) − h B ( y + e μ )
(iii) With S y μ = ∑ x ∈ B ( y ) , o μ = 0 φ ( x , μ ) S^\mu_y=\sum_{x\in B(y),\,o_\mu=0}\varphi_{(x,\mu)} S y μ = ∑ x ∈ B ( y ) , o μ = 0 φ ( x , μ )
V y , μ ( φ ) = 1 8 ∑ x ∈ B ( y ) o μ = 1 φ ( x , μ ) + 1 16 ( S y μ + S y + e μ μ ) . V_{y,\mu}(\varphi)=\frac{1}{8}\sum_{\substack{x\in B(y)\\o_\mu=1}}\varphi_{(x,\mu)}+\frac{1}{16}\left(S^\mu_y+S^\mu_{y+e_\mu}\right). V y , μ ( φ ) = 8 1 x ∈ B ( y ) o μ = 1 ∑ φ ( x , μ ) + 16 1 ( S y μ + S y + e μ μ ) . Proof. (i) A tree link of direction μ \mu μ o μ = 0 o_\mu=0 o μ = 0 2 μ 2^\mu 2 μ μ \mu μ o 3 = 0 o_3=0 o 3 = 0 o 2 = 0 o_2=0 o 2 = 0 o μ = 0 o_\mu=0 o μ = 0 h h h x ∈ B ( y ) x\in B(y) x ∈ B ( y ) x + 2 e μ ∈ B ( y + e μ ) x+2e_\mu\in B(y+e_\mu) x + 2 e μ ∈ B ( y + e μ ) d h dh d h h ( x ) − h ( x + 2 e μ ) h(x)-h(x+2e_\mu) h ( x ) − h ( x + 2 e μ ) x x x ( x , μ ) (x,\mu) ( x , μ ) ( x + e μ , μ ) (x+e_\mu,\mu) ( x + e μ , μ ) o μ = 1 o_\mu=1 o μ = 1 S y + e μ μ S^\mu_{y+e_\mu} S y + e μ μ o μ = 1 o_\mu=1 o μ = 1
By (ii) the residual gauge freedom is the coarse gauge group, which acts on the block averages as the links of the coarse lattice. Let W W W Φ W \Phi_W Φ W φ \varphi φ ∪ W \cup W ∪ W V y , μ ( φ ) = 0 V_{y,\mu}(\varphi)=0 V y , μ ( φ ) = 0 y ∈ W y\in W y ∈ W μ \mu μ Q ( φ ) = ∑ p ( d φ ) p 2 Q(\varphi)=\sum_p(d\varphi)_p^2 Q ( φ ) = ∑ p ( d φ ) p 2 φ \varphi φ
Theorem 85. *For every set of blocks W W W φ ∈ Φ W \varphi\in\Phi_W φ ∈ Φ W
∑ e φ e 2 ≤ 52 Q ( φ ) . \sum_e\varphi_e^2\le52Q(\varphi). e ∑ φ e 2 ≤ 52 Q ( φ ) . Proof. Inside a block. Let e = ( x , μ ) e=(x,\mu) e = ( x , μ ) o μ = 0 o_\mu=0 o μ = 0 ν 1 > ⋯ > ν k \nu_1>\cdots>\nu_k ν 1 > ⋯ > ν k μ \mu μ o ν = 1 o_\nu=1 o ν = 1 1 ≤ k < 3 1\le k<3 1 ≤ k < 3 x 0 = x x_0=x x 0 = x x i = x i − 1 − e ν i x_i=x_{i-1}-e_{\nu_i} x i = x i − 1 − e ν i p i p_i p i x i x_i x i { μ , ν i } \{\mu,\nu_i\} { μ , ν i } ( x i , μ ) (x_i,\mu) ( x i , μ ) ( x i − 1 , μ ) (x_{i-1},\mu) ( x i − 1 , μ ) ( x i , ν i ) (x_i,\nu_i) ( x i , ν i ) ( x i + e μ , ν i ) (x_i+e_\mu,\nu_i) ( x i + e μ , ν i ) ν i \nu_i ν i ( x k , μ ) (x_k,\mu) ( x k , μ ) φ e = ± ( d φ ) p 1 ± ⋯ ± ( d φ ) p k \varphi_e=\pm(d\varphi)_{p_1}\pm\cdots\pm(d\varphi)_{p_k} φ e = ± ( d φ ) p 1 ± ⋯ ± ( d φ ) p k φ e 2 ≤ k ∑ i ( d φ ) p i 2 \varphi_e^2\le k\sum_i(d\varphi)_{p_i}^2 φ e 2 ≤ k ∑ i ( d φ ) p i 2 x ′ x' x ′ { μ , ν } \{\mu,\nu\} { μ , ν } ( x ′ + e ν + ∑ ρ ∈ R e ρ , μ ) (x'+e_\nu+\sum_{\rho\in R}e_\rho,\mu) ( x ′ + e ν + ∑ ρ ∈ R e ρ , μ ) R ⊂ { ν + 1 , … , 3 } R\subset\{\nu+1,\ldots,3\} R ⊂ { ν + 1 , … , 3 } k = c + 1 + ∣ R ∣ k=c+1+|R| k = c + 1 + ∣ R ∣ c c c x ′ x' x ′ μ \mu μ ν \nu ν 2 3 − ( c + 1 ) + ( 3 − ν ) 2 2 − c 2^{3-(c+1)+(3-\nu)}2^{2-c} 2 3 − ( c + 1 ) + ( 3 − ν ) 2 2 − c μ = 0 \mu=0 μ = 0 ν = 1 \nu=1 ν = 1 φ e 2 \varphi_e^2 φ e 2 8 Q i n 8Q_{\mathrm{in}} 8 Q in Q i n Q_{\mathrm{in}} Q in
Between blocks. Fix y ∈ W y\in W y ∈ W μ \mu μ ( x , μ ) (x,\mu) ( x , μ ) x ∈ B ( y ) x\in B(y) x ∈ B ( y ) o μ = 1 o_\mu=1 o μ = 1 a ∈ { 0 , 1 } 3 a\in\{0,1\}^3 a ∈ { 0 , 1 } 3 a a a a ′ a' a ′ ν \nu ν o ν = 0 o_\nu=0 o ν = 0 a a a x x x { μ , ν } \{\mu,\nu\} { μ , ν } ( x , ν ) (x,\nu) ( x , ν ) B ( y ) B(y) B ( y ) ( x + e μ , ν ) (x+e_\mu,\nu) ( x + e μ , ν ) B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) φ a − φ a ′ = ± ( d φ ) p ± ψ ± ψ ′ \varphi_a-\varphi_{a'}=\pm(d\varphi)_p\pm\psi\pm\psi' φ a − φ a ′ = ± ( d φ ) p ± ψ ± ψ ′ ψ , ψ ′ \psi,\psi' ψ , ψ ′ { 0 , 1 } 3 \{0,1\}^3 { 0 , 1 } 3 V y , μ ( φ ) = 0 V_{y,\mu}(\varphi)=0 V y , μ ( φ ) = 0 φ a \varphi_a φ a m = − ( S y μ + S y + e μ μ ) / 16 m=-(S_y^\mu+S_{y+e_\mu}^\mu)/16 m = − ( S y μ + S y + e μ μ ) /16 8 m 2 ≤ 1 2 8m^2\leq\frac12 8 m 2 ≤ 2 1 S y μ S_y^\mu S y μ S y + e μ μ S_{y+e_\mu}^\mu S y + e μ μ { 0 , 1 } 3 \{0,1\}^3 { 0 , 1 } 3 2 2 2
∑ a φ a 2 = 8 m 2 + ∑ a ( φ a − m ) 2 ≤ 8 m 2 + 3 2 ∑ e d g e s ( ( d φ ) p 2 + ψ 2 + ψ ′ 2 ) . \sum_a\varphi_a^2=8m^2+\sum_a(\varphi_a-m)^2
\leq8m^2+\frac{3}{2}\sum_{\mathrm{edges}}\left((d\varphi)_p^2+\psi^2+\psi'^2\right). a ∑ φ a 2 = 8 m 2 + a ∑ ( φ a − m ) 2 ≤ 8 m 2 + 2 3 edges ∑ ( ( d φ ) p 2 + ψ 2 + ψ ′2 ) . Summed over y ∈ W y\in W y ∈ W μ \mu μ ( z , ν ) (z,\nu) ( z , ν ) ψ \psi ψ ψ ′ \psi' ψ ′ μ ≠ ν \mu\neq\nu μ = ν ψ \psi ψ o μ = 1 o_\mu=1 o μ = 1 ψ ′ \psi' ψ ′ o μ = 0 o_\mu=0 o μ = 0 S ν S^\nu S ν 3 ⋅ 3 2 + 2 ⋅ 1 2 = 11 2 3\cdot\frac{3}{2}+2\cdot\frac{1}{2}=\frac{11}{2} 3 ⋅ 2 3 + 2 ⋅ 2 1 = 2 11 3 2 Q f a c e + 11 2 I \frac{3}{2}Q_{\mathrm{face}}+\frac{11}{2}I 2 3 Q face + 2 11 I
∑ e φ e 2 ≤ 13 2 I + 3 2 Q f a c e ≤ 52 Q i n + 3 2 Q f a c e ≤ 52 Q ( φ ) . \sum_e\varphi_e^2\leq\frac{13}{2}I+\frac{3}{2}Q_{\mathrm{face}}\leq52Q_{\mathrm{in}}+\frac{3}{2}Q_{\mathrm{face}}\leq52Q(\varphi). e ∑ φ e 2 ≤ 2 13 I + 2 3 Q face ≤ 52 Q in + 2 3 Q face ≤ 52 Q ( φ ) . With W W W ℓ 2 \ell^2 ℓ 2 Z 4 \mathbb{Z}^4 Z 4 1 52 \frac{1}{52} 52 1 0.10748776948684 … 0.10748776948684\ldots 0.10748776948684 … ( 0 , π , π , π ) (0,\pi,\pi,\pi) ( 0 , π , π , π ) [53 ] and reproduced in the archive; the theorem does not use it. The cube conditions now take the following form. Let n n n V V V [ 0 , n ) 4 [0,n)^4 [ 0 , n ) 4 W W W η \eta η v = ( v y , μ ) y ∈ W v=(v_{y,\mu})_{y\in W} v = ( v y , μ ) y ∈ W γ V ( ⋅ ∣ η , v ) \gamma_V(\mathord\cdot\mid\eta,v) γ V ( ⋅ ∣ η , v ) R V \mathbb{R}^V R V exp ( − b 2 ∑ p ∩ V ≠ ∅ ( d φ ) p 2 ) \exp\left(-\frac{b}{2}\sum_{p\cap V\neq\varnothing}(d\varphi)_p^2\right) exp ( − 2 b ∑ p ∩ V = ∅ ( d φ ) p 2 ) { V y , μ ( φ ) = v y , μ } \{V_{y,\mu}(\varphi)=v_{y,\mu}\} { V y , μ ( φ ) = v y , μ } φ = η \varphi=\eta φ = η V V V C V , j c C^c_{V,j} C V , j c D c ( n , α ) D_c(n,\alpha) D c ( n , α ) C V , j C_{V,j} C V , j K c K_c K c D ( n , α ) D(n,\alpha) D ( n , α ) v v v 49 49 49
Lemma 86. Let H ≥ 0 H\geq0 H ≥ 0 R m \mathbb{R}^m R m ∥ H ∥ ≤ h \lVert H\rVert\leq h ∥ H ∥ ≤ h B : R m → R k B:\mathbb{R}^m\to\mathbb{R}^k B : R m → R k B B ⊤ ≥ β > 0 BB^\top\geq\beta>0 B B ⊤ ≥ β > 0 ⟨ x , H x ⟩ ≥ a ∣ x ∣ 2 \langle x,Hx\rangle\geq a|x|^2 ⟨ x , H x ⟩ ≥ a ∣ x ∣ 2 ker B \ker B ker B s > 0 s>0 s > 0 K = ( H s B ⊤ s B 0 ) K=\begin{pmatrix}H&sB^\top\\sB&0\end{pmatrix} K = ( H s B s B ⊤ 0 ) K ( x , y ) = ( f , g ) K(x,y)=(f,g) K ( x , y ) = ( f , g ) ∣ x ∣ ≤ ( ∣ f ∣ + h ∣ x 1 ∣ ) / a + ∣ x 1 ∣ |x|\leq(|f|+h|x_1|)/a+|x_1| ∣ x ∣ ≤ ( ∣ f ∣ + h ∣ x 1 ∣ ) / a + ∣ x 1 ∣ ∣ x 1 ∣ ≤ ∣ g ∣ / ( s β ) |x_1|\leq|g|/(s\sqrt{\beta}) ∣ x 1 ∣ ≤ ∣ g ∣/ ( s β ) ∣ y ∣ ≤ ( ∣ f ∣ + h ∣ x ∣ ) / ( s β ) |y|\leq(|f|+h|x|)/(s\sqrt{\beta}) ∣ y ∣ ≤ ( ∣ f ∣ + h ∣ x ∣ ) / ( s β )
Proof. Put x 1 = B ⊤ ( B B ⊤ ) − 1 g / s x_1=B^\top(BB^\top)^{-1}g/s x 1 = B ⊤ ( B B ⊤ ) − 1 g / s s B x 1 = g sBx_1=g s B x 1 = g ∣ x 1 ∣ ≤ ∣ g ∣ / ( s β ) |x_1|\leq|g|/(s\sqrt{\beta}) ∣ x 1 ∣ ≤ ∣ g ∣/ ( s β ) ker B \ker B ker B H H H a a a x 0 ∈ ker B x_0\in\ker B x 0 ∈ ker B H x 0 − f + H x 1 ⊥ ker B Hx_0-f+Hx_1\perp\ker B H x 0 − f + H x 1 ⊥ ker B a ∣ x 0 ∣ ≤ ∣ f ∣ + h ∣ x 1 ∣ a|x_0|\leq|f|+h|x_1| a ∣ x 0 ∣ ≤ ∣ f ∣ + h ∣ x 1 ∣ f − H ( x 0 + x 1 ) f-H(x_0+x_1) f − H ( x 0 + x 1 ) ( ker B ) ⊥ = ran B ⊤ (\ker B)^\perp=\operatorname{ran}B^\top ( ker B ) ⊥ = ran B ⊤ y = ( B B ⊤ ) − 1 B ( f − H x ) / s y=(BB^\top)^{-1}B(f-Hx)/s y = ( B B ⊤ ) − 1 B ( f − H x ) / s s B ⊤ y = f − H x sB^\top y=f-Hx s B ⊤ y = f − H x f = g = 0 f=g=0 f = g = 0 K K K
Lemma 87 (Combes–Thomas [27 ] ). Let K K K X X X ∥ K − 1 ∥ ≤ M 0 \lVert K^{-1}\rVert\leq M_0 ∥ K − 1 ∥ ≤ M 0 ∑ b ∣ K a b ∣ ≤ S \sum_b|K_{ab}|\leq S ∑ b ∣ K ab ∣ ≤ S a a a d : X → R d:X\to\mathbb{R} d : X → R ∣ d ( a ) − d ( b ) ∣ ≤ 1 |d(a)-d(b)|\leq1 ∣ d ( a ) − d ( b ) ∣ ≤ 1 K a b ≠ 0 K_{ab}\neq0 K ab = 0 ( e λ − 1 ) S M 0 ≤ 1 2 (e^\lambda-1)SM_0\leq\frac12 ( e λ − 1 ) S M 0 ≤ 2 1 ∥ e λ d K − 1 e − λ d ∥ ≤ 2 M 0 \lVert e^{\lambda d}K^{-1}e^{-\lambda d}\rVert\leq2M_0 ∥ e λ d K − 1 e − λ d ∥ ≤ 2 M 0
Proof. e λ d K e − λ d = K + E e^{\lambda d}Ke^{-\lambda d}=K+E e λ d K e − λ d = K + E ∣ E a b ∣ ≤ ( e λ − 1 ) ∣ K a b ∣ |E_{ab}|\leq(e^\lambda-1)|K_{ab}| ∣ E ab ∣ ≤ ( e λ − 1 ) ∣ K ab ∣ ∥ E ∥ ≤ ( e λ − 1 ) S \lVert E\rVert\leq(e^\lambda-1)S ∥ E ∥ ≤ ( e λ − 1 ) S ∥ K − 1 E ∥ ≤ 1 2 \lVert K^{-1}E\rVert\leq\frac12 ∥ K − 1 E ∥ ≤ 2 1 ( K + E ) − 1 = ( 1 + K − 1 E ) − 1 K − 1 (K+E)^{-1}=(1+K^{-1}E)^{-1}K^{-1} ( K + E ) − 1 = ( 1 + K − 1 E ) − 1 K − 1 2 M 0 2M_0 2 M 0
Theorem 88. For every b > 0 b>0 b > 0 n n n j j j
(i) γ V ( ⋅ ∣ η , v ) \gamma_V(\mathord\cdot\mid\eta,v) γ V ( ⋅ ∣ η , v ) η \eta η v v v ( η , v ) (\eta,v) ( η , v ) C V , j c = ∥ u j ∥ 1 C^c_{V,j}=\lVert u_j\rVert_1 C V , j c = ∥ u j ∥ 1 u j ∈ R V u_j\in\mathbb{R}^V u j ∈ R V η j \eta_j η j u j u_j u j b b b
(ii) ∑ e ∈ V e 2 λ 0 d ( e , j ) u j ( e ) 2 ≤ 3500 2 \sum_{e\in V}e^{2\lambda_0d(e,j)}u_j(e)^2\leq3500^2 ∑ e ∈ V e 2 λ 0 d ( e , j ) u j ( e ) 2 ≤ 350 0 2 ∥ u j ∥ 1 ≤ 10 14 \lVert u_j\rVert_1\leq10^{14} ∥ u j ∥ 1 ≤ 1 0 14 λ 0 = 2.4 ⋅ 10 − 5 \lambda_0=2.4\cdot10^{-5} λ 0 = 2.4 ⋅ 1 0 − 5 d ( e ) d(e) d ( e ) ℓ ∞ \ell^\infty ℓ ∞ e e e j j j
(iii) D c ( n , α ) D_c(n,\alpha) D c ( n , α ) α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 ) n ≥ 2 ⋅ 10 15 / α n\geq2\cdot10^{15}/\alpha n ≥ 2 ⋅ 1 0 15 / α b > 0 b>0 b > 0
Proof. (i) Let H H H ∑ p ∩ V ≠ ∅ ( d φ ) p 2 \sum_{p\cap V\neq\varnothing}(d\varphi)_p^2 ∑ p ∩ V = ∅ ( d φ ) p 2 R V \mathbb{R}^V R V B B B V y , μ V_{y,\mu} V y , μ y ∈ W y\in W y ∈ W V V V W W W H ≥ 1 52 H\geq\frac{1}{52} H ≥ 52 1 ker B \ker B ker B b − 1 b^{-1} b − 1 H H H ker B \ker B ker B m m m ( η , v ) (\eta,v) ( η , v ) j j j u j u_j u j Q ( u + δ j ) Q(u+\delta_j) Q ( u + δ j ) B u = g Bu=g B u = g g g g j j j C V , j c = ∥ u j ∥ 1 C^c_{V,j}=\lVert u_j\rVert_1 C V , j c = ∥ u j ∥ 1
(ii) The pair ( u j , y ) (u_j,y) ( u j , y ) y y y K ( u j , y ) = ( f , g ) K(u_j,y)=(f,g) K ( u j , y ) = ( f , g ) K K K s = 96 s=96 s = 96 f = − ∑ p ∋ j ( d δ j ) p d p ∣ V f=-\sum_{p\ni j}(d\delta_j)_p d_p|_V f = − ∑ p ∋ j ( d δ j ) p d p ∣ V d p ∣ V d_p|_V d p ∣ V p p p V V V ∣ H ∣ |H| ∣ H ∣ 24 24 24 h ≤ 24 h\leq24 h ≤ 24 ∣ f ∣ ≤ 6 3 |f|\leq6\sqrt{3} ∣ f ∣ ≤ 6 3 ∣ g ∣ ≤ 1 16 |g|\leq\frac{1}{16} ∣ g ∣ ≤ 16 1 W W W 1 16 \frac{1}{16} 16 1 8 ⋅ 1 64 = 1 8 8\cdot\frac{1}{64}=\frac18 8 ⋅ 64 1 = 8 1 ( y , μ ) (y,\mu) ( y , μ ) ( y ± e μ , μ ) (y\pm e_\mu,\mu) ( y ± e μ , μ ) 8 ⋅ 1 256 = 1 32 8\cdot\frac{1}{256}=\frac{1}{32} 8 ⋅ 256 1 = 32 1 B B ⊤ ≥ 1 256 BB^\top\geq\frac{1}{256} B B ⊤ ≥ 256 1 a = 1 52 a=\frac{1}{52} a = 52 1 ∥ K − 1 ∥ ≤ M 0 = 105 \lVert K^{-1}\rVert\leq M_0=105 ∥ K − 1 ∥ ≤ M 0 = 105 K K K S = max ( 24 + 96 8 , 96 ⋅ 2 ) = 192 S=\max(24+\frac{96}{8},96\cdot2)=192 S = max ( 24 + 8 96 , 96 ⋅ 2 ) = 192 1 8 \frac18 8 1 1 16 \frac{1}{16} 16 1 1 8 \frac18 8 1 B B B 2 y + 1 2 ( 1 , 1 , 1 , 1 ) + e μ 2y+\frac12(1,1,1,1)+e_\mu 2 y + 2 1 ( 1 , 1 , 1 , 1 ) + e μ d d d ℓ ∞ \ell^\infty ℓ ∞ j j j λ = log ( 1 + 1 / 40320 ) > λ 0 \lambda=\log(1+1/40320)>\lambda_0 λ = log ( 1 + 1/40320 ) > λ 0
∑ e e 2 λ d ( e ) u j ( e ) 2 ≤ ( 2 M 0 e λ ( ∣ f ∣ + s ∣ g ∣ ) ) 2 ≤ 3500 2 , \sum_e e^{2\lambda d(e)}u_j(e)^2\leq\left(2M_0e^\lambda(|f|+s|g|)\right)^2\leq3500^2, e ∑ e 2 λ d ( e ) u j ( e ) 2 ≤ ( 2 M 0 e λ ( ∣ f ∣ + s ∣ g ∣ ) ) 2 ≤ 350 0 2 , since f f f g g g j j j 4 ( 64 k 3 + 16 k ) 4(64k^3+16k) 4 ( 64 k 3 + 16 k ) ℓ ∞ \ell^\infty ℓ ∞ k ≥ 1 k\geq1 k ≥ 1 j j j k − 1 k-1 k − 1
∥ u j ∥ 1 2 ≤ 3500 2 ⋅ 4 e 2 λ ( 1 + ∑ k ≥ 1 ( 64 k 3 + 16 k ) e − 2 λ k ) ≤ ( 10 14 ) 2 . (19) \lVert u_j\rVert_1^2\leq3500^2\cdot4e^{2\lambda}\left(1+\sum_{k\geq1}(64k^3+16k)e^{-2\lambda k}\right)\leq(10^{14})^2. \tag*{(19)} ∥ u j ∥ 1 2 ≤ 350 0 2 ⋅ 4 e 2 λ ( 1 + k ≥ 1 ∑ ( 64 k 3 + 16 k ) e − 2 λk ) ≤ ( 1 0 14 ) 2 . (19) (iii) C V x , j c = 0 C^c_{V_x,j}=0 C V x , j c = 0 j j j V x V_x V x V x V_x V x z z z j j j x + [ − 1 , n + 1 ] 4 x+[-1,n+1]^4 x + [ − 1 , n + 1 ] 4 z ∉ x + [ 0 , n ) 4 z\notin x+[0,n)^4 z ∈ / x + [ 0 , n ) 4 j ∉ V x j\notin V_x j ∈ / V x n 2 + 2 \frac{n}{2}+2 2 n + 2 x i x_i x i n 2 \frac{n}{2} 2 n
K c c ≤ ( ( n 2 + 2 ) 4 − ( n 2 ) 4 ) 10 14 ≤ 17 n 10 14 N n K^c_c \leq\left(\left(\frac{n}{2}+2\right)^4-\left(\frac{n}{2}\right)^4\right)10^{14}\leq\frac{17}{n}10^{14}N_n K c c ≤ ( ( 2 n + 2 ) 4 − ( 2 n ) 4 ) 1 0 14 ≤ n 17 1 0 14 N n for n ≥ 10 4 n\geq10^4 n ≥ 1 0 4 α N n \alpha N_n α N n n ≥ 2 ⋅ 10 15 / α n\geq2\cdot10^{15}/\alpha n ≥ 2 ⋅ 1 0 15 / α
The constants are crude. The computations of [53 ] give C V , j c ≤ 12.48 C^c_{V,j}\leq12.48 C V , j c ≤ 12.48 n = 4 n=4 n = 4 K c c / N n = a c / n + O ( n − 2 ) K^c_c/N_n=a_c/n+O(n^{-2}) K c c / N n = a c / n + O ( n − 2 ) max c a c = 61.18 \max_c a_c=61.18 max c a c = 61.18 D c ( n , 0.95 ) D_c(n,0.95) D c ( n , 0.95 ) n = 64 n=64 n = 64 n ≤ 4 n\leq4 n ≤ 4 ∑ j ∈ c ∖ V ∑ c ′ w c ′ C V , j c ′ ≤ α N n w c \sum_{j\in c\setminus V}\sum_{c'}w_{c'}C^{c'}_{V,j}\leq\alpha N_nw_c ∑ j ∈ c ∖ V ∑ c ′ w c ′ C V , j c ′ ≤ α N n w c w w w C V , j c ′ C^{c'}_{V,j} C V , j c ′ ∥ u j ∥ 1 \lVert u_j\rVert_1 ∥ u j ∥ 1 c ′ c' c ′ α N n \alpha N_n α N n min c K c c \min_c K^c_c min c K c c [53 ] .
The block averages can be integrated out one scale at a time. For a field v v v Q 1 ( v ) = ∑ P ( d v ) P 2 Q_1(v)=\sum_P(dv)_P^2 Q 1 ( v ) = ∑ P ( d v ) P 2 P P P V ( φ ) = ( V y , μ ( φ ) ) y , μ V(\varphi)=(V_{y,\mu}(\varphi))_{y,\mu} V ( φ ) = ( V y , μ ( φ ) ) y , μ V ( k ) , Q k V^{(k)},Q_k V ( k ) , Q k k k k 2 k 2^k 2 k
Proposition 89. Let φ \varphi φ Z 4 \mathbb{Z}^4 Z 4
(i) For the coarse plaquette P P P y y y { μ , ν } \{\mu,\nu\} { μ , ν } 16 ( d V ( φ ) ) P 16(dV(\varphi))_P 16 ( d V ( φ ) ) P x ∈ B ( y ) x\in B(y) x ∈ B ( y ) x x x x + 2 e μ x+2e_\mu x + 2 e μ x + 2 e ν x+2e_\nu x + 2 e ν Z 4 \mathbb{Z}^4 Z 4 Q 1 ( V ( φ ) ) ≤ Q ( φ ) Q_1(V(\varphi))\leq Q(\varphi) Q 1 ( V ( φ )) ≤ Q ( φ ) Q k ( V ( k ) ( φ ) ) ≤ Q ( φ ) Q_k(V^{(k)}(\varphi))\leq Q(\varphi) Q k ( V ( k ) ( φ )) ≤ Q ( φ ) k ≥ 1 k\geq1 k ≥ 1
(ii) For every finitely supported v v v φ ∗ ( v ) \varphi^*(v) φ ∗ ( v ) v y , μ v_{y,\mu} v y , μ ( x , μ ) (x,\mu) ( x , μ ) x ∈ B ( y ) x\in B(y) x ∈ B ( y ) o μ = 1 o_\mu=1 o μ = 1 V ( φ ∗ ( v ) ) = v V(\varphi^*(v))=v V ( φ ∗ ( v )) = v Q ( φ ∗ ( v ) ) = 4 Q 1 ( v ) Q(\varphi^*(v))=4Q_1(v) Q ( φ ∗ ( v )) = 4 Q 1 ( v )
(iii) The effective form S 1 ( v ) = inf { Q ( φ ) : V ( φ ) = v } S_1(v)=\inf\{Q(\varphi):V(\varphi)=v\} S 1 ( v ) = inf { Q ( φ ) : V ( φ ) = v } Q 1 ≤ S 1 ≤ 4 Q 1 Q_1\leq S_1\leq4Q_1 Q 1 ≤ S 1 ≤ 4 Q 1 S k ( w ) = inf { Q ( φ ) : V ( k ) ( φ ) = w } ≥ Q k ( w ) S_k(w)=\inf\{Q(\varphi):V^{(k)}(\varphi)=w\}\geq Q_k(w) S k ( w ) = inf { Q ( φ ) : V ( k ) ( φ ) = w } ≥ Q k ( w ) k k k
(iv) For every k k k W W W 2 k 2^k 2 k w w w Φ W \Phi_W Φ W ∑ e w e 2 ≤ 52 S k ( w ) \sum_e w_e^2\leq52S_k(w) ∑ e w e 2 ≤ 52 S k ( w )
Proof. (i) The path of length two from x ∈ B ( y ) x\in B(y) x ∈ B ( y ) μ \mu μ x + 2 e μ ∈ B ( y + e μ ) x+2e_\mu\in B(y+e_\mu) x + 2 e μ ∈ B ( y + e μ ) ( d V ) P (dV)_P ( d V ) P φ \varphi φ z z z { μ , ν } \{\mu,\nu\} { μ , ν } x x x z − x ∈ { 0 , 1 } e μ + { 0 , 1 } e ν z-x\in\{0,1\}e_\mu+\{0,1\}e_\nu z − x ∈ { 0 , 1 } e μ + { 0 , 1 } e ν x x x ( d V ) P 2 ≤ 1 16 ∑ x L x 2 ≤ 1 4 ∑ x ( d φ ) P x 2 (dV)_P^2\leq\frac{1}{16}\sum_x\mathcal{L}_x^2\leq\frac{1}{4}\sum_x(d\varphi)_{P_x}^2 ( d V ) P 2 ≤ 16 1 ∑ x L x 2 ≤ 4 1 ∑ x ( d φ ) P x 2 L x \mathcal{L}_x L x x x x P P P Q 1 ( V ( φ ) ) ≤ Q ( φ ) Q_1(V(\varphi))\leq Q(\varphi) Q 1 ( V ( φ )) ≤ Q ( φ ) V ( k ) V^{(k)} V ( k )
(ii) The tree links have o μ = 0 o_\mu=0 o μ = 0 φ ∗ \varphi^* φ ∗ 1 8 ⋅ 8 v y , μ \frac{1}{8}\cdot8v_{y,\mu} 8 1 ⋅ 8 v y , μ o μ = 0 o_\mu=0 o μ = 0 μ \mu μ x x x { μ , ν } \{\mu,\nu\} { μ , ν } o μ = 1 o_\mu=1 o μ = 1 o ν = 0 o_\nu=0 o ν = 0 ( x , μ ) (x,\mu) ( x , μ ) ( x + e ν , μ ) (x+e_\nu,\mu) ( x + e ν , μ ) v y , μ v_{y,\mu} v y , μ ν \nu ν o μ = 0 o_\mu=0 o μ = 0 o ν = 1 o_\nu=1 o ν = 1 o μ = o ν = 1 o_\mu=o_\nu=1 o μ = o ν = 1 v y , μ v_{y,\mu} v y , μ v y + e μ , ν v_{y+e_\mu,\nu} v y + e μ , ν v y + e ν , μ v_{y+e_\nu,\mu} v y + e ν , μ v y , ν v_{y,\nu} v y , ν ( d v ) P (dv)_P ( d v ) P P P P
(iii) The lower bound is (i) and the upper bound is (ii); for S k S_k S k k k k
(iv) Theorem 85 holds on every lattice, so ∑ e w e 2 ≤ 52 Q k ( w ) ≤ 52 S k ( w ) \sum_e w_e^2\leq52Q_k(w)\leq52S_k(w) ∑ e w e 2 ≤ 52 Q k ( w ) ≤ 52 S k ( w )
In the Bloch decomposition the eigenvalues of S 1 S_1 S 1 Q 1 Q_1 Q 1 [ 1 , 4 ] [1,4] [ 1 , 4 ] ( π , π , 0 , 0 ) (\pi,\pi,0,0) ( π , π , 0 , 0 ) g 2 g^2 g 2 S k S_k S k S k S_k S k
Lemma 90. Let ( X , d X ) (X,d_X) ( X , d X ) ( Y , d Y ) (Y,d_Y) ( Y , d Y ) ρ , ρ ′ \rho,\rho' ρ , ρ ′ Y Y Y κ , κ ′ \kappa,\kappa' κ , κ ′ Y Y Y X X X W 1 ( κ ( ⋅ ∣ y ) , κ ′ ( ⋅ ∣ y ) ) ≤ ϵ W_1(\kappa(\cdot\mid y),\kappa'(\cdot\mid y))\leq\epsilon W 1 ( κ ( ⋅ ∣ y ) , κ ′ ( ⋅ ∣ y )) ≤ ϵ W 1 ( κ ′ ( ⋅ ∣ y ) , κ ′ ( ⋅ ∣ y ′ ) ) ≤ d Y ( y , y ′ ) W_1(\kappa'(\cdot\mid y),\kappa'(\cdot\mid y'))\leq d_Y(y,y') W 1 ( κ ′ ( ⋅ ∣ y ) , κ ′ ( ⋅ ∣ y ′ )) ≤ d Y ( y , y ′ ) y , y ′ y,y' y , y ′ π = ∫ κ ( ⋅ ∣ y ) d ρ ( y ) \pi=\int\kappa(\cdot\mid y)\,d\rho(y) π = ∫ κ ( ⋅ ∣ y ) d ρ ( y ) π ′ = ∫ κ ′ ( ⋅ ∣ y ) d ρ ′ ( y ) \pi'=\int\kappa'(\cdot\mid y)\,d\rho'(y) π ′ = ∫ κ ′ ( ⋅ ∣ y ) d ρ ′ ( y )
W 1 ( π , π ′ ) ≤ ϵ + W 1 d Y ( ρ , ρ ′ ) . W_1(\pi,\pi')\leq\epsilon+W_1^{d_Y}(\rho,\rho'). W 1 ( π , π ′ ) ≤ ϵ + W 1 d Y ( ρ , ρ ′ ) . Proof. Let F F F X X X κ F ( v ) = ∫ F d κ ( ⋅ ∣ v ) \kappa F(v)=\int F\,d\kappa(\cdot\mid v) κ F ( v ) = ∫ F d κ ( ⋅ ∣ v ) ∫ F d π − ∫ F d π ′ = ∫ ( κ F − κ ′ F ) d ρ + ( ∫ κ ′ F d ρ − ∫ κ ′ F d ρ ′ ) \int F\,d\pi-\int F\,d\pi'=\int(\kappa F-\kappa'F)\,d\rho+\left(\int\kappa'F\,d\rho-\int\kappa'F\,d\rho'\right) ∫ F d π − ∫ F d π ′ = ∫ ( κ F − κ ′ F ) d ρ + ( ∫ κ ′ F d ρ − ∫ κ ′ F d ρ ′ ) ϵ \epsilon ϵ κ ′ F \kappa'F κ ′ F d Y d_Y d Y W 1 d Y ( ρ , ρ ′ ) W_1^{d_Y}(\rho,\rho') W 1 d Y ( ρ , ρ ′ )
A law γ \gamma γ R V \mathbb{R}^V R V γ ( ⋅ ∣ η ) = ∫ γ ( ⋅ ∣ η , ν ) ρ V ( d ν ∣ η ) \gamma(\cdot\mid\eta)=\int\gamma(\cdot\mid\eta,\nu)\,\rho_V(d\nu\mid\eta) γ ( ⋅ ∣ η ) = ∫ γ ( ⋅ ∣ η , ν ) ρ V ( d ν ∣ η ) ρ V ( ⋅ ∣ η ) \rho_V(\cdot\mid\eta) ρ V ( ⋅ ∣ η ) ( V y , μ ) y ∈ W (V_{y,\mu})_{y\in W} ( V y , μ ) y ∈ W exp ( − β 2 ∑ P ( d φ ) P 2 ) g ( V ( φ ) ) \exp\left(-\frac{\beta}{2}\sum_P(d\varphi)_P^2\right)g(V(\varphi)) exp ( − 2 β ∑ P ( d φ ) P 2 ) g ( V ( φ )) g g g
Corollary 91. Let r w ∈ R V r_w \in\mathbb{R}^V r w ∈ R V ∑ p ∩ V ≠ ∅ ( d φ ) 2 \sum_{p\cap V\ne\varnothing}(d\varphi)^2 ∑ p ∩ V = ∅ ( d φ ) 2 V y , y ′ ( r w ) = δ w , ( y , μ ′ ) V_{y,y'}(r_w)=\delta_{w,(y,\mu')} V y , y ′ ( r w ) = δ w , ( y , μ ′ ) ( y , μ ′ ) , y ′ ∈ W (y,\mu'),y'\in W ( y , μ ′ ) , y ′ ∈ W L w = ∥ r w ∥ 1 L_w=\|r_w\|_1 L w = ∥ r w ∥ 1 d L ( v , v ′ ) = ∑ w L w ∣ v w − v w ′ ∣ d_L(v,v')=\sum_w L_w|v_w-v'_w| d L ( v , v ′ ) = ∑ w L w ∣ v w − v w ′ ∣
C V , j ≤ C V , j c + C V , j ρ , C V , j ρ = sup η , η ′ W 1 d L ( ρ V ( ⋅ ∣ η ) , ρ V ( ⋅ ∣ η ′ ) ) ∣ η j − η j ′ ∣ , C_{V,j}\le C^c_{V,j}+C^\rho_{V,j},\qquad C^\rho_{V,j}=\sup_{\eta,\eta'}\frac{W^{d_L}_1(\rho_V(\cdot\mid\eta),\rho_V(\cdot\mid\eta'))}{|\eta_j-\eta'_j|}, C V , j ≤ C V , j c + C V , j ρ , C V , j ρ = η , η ′ sup ∣ η j − η j ′ ∣ W 1 d L ( ρ V ( ⋅ ∣ η ) , ρ V ( ⋅ ∣ η ′ )) , the supremum over exteriors that differ only at j j j L w ≤ 4 ⋅ 10 14 L_w\le4\cdot10^{14} L w ≤ 4 ⋅ 1 0 14 K c ≤ K c c + K c ρ K_c\le K^c_c+K^\rho_c K c ≤ K c c + K c ρ K c ρ = ∑ j ∈ c ∖ V C V , j ρ K^\rho_c=\sum_{j\in c\setminus V}C^\rho_{V,j} K c ρ = ∑ j ∈ c ∖ V C V , j ρ K c ρ ≤ α 2 N n K^\rho_c\le\frac{\alpha}{2}N_n K c ρ ≤ 2 α N n n ≥ 4 ⋅ 10 15 / α n\ge4\cdot10^{15}/\alpha n ≥ 4 ⋅ 1 0 15 / α D ( n , α ) D(n,\alpha) D ( n , α )
Proof. Apply Lemma 90 with κ = γ V c ( ⋅ ∣ η , ⋅ ) \kappa=\gamma^c_V(\cdot\mid\eta,\cdot) κ = γ V c ( ⋅ ∣ η , ⋅ ) κ ′ = γ V c ( ⋅ ∣ η ′ , ⋅ ) \kappa'=\gamma^c_V(\cdot\mid\eta',\cdot) κ ′ = γ V c ( ⋅ ∣ η ′ , ⋅ ) ρ = ρ V ( ⋅ ∣ η ) \rho=\rho_V(\cdot\mid\eta) ρ = ρ V ( ⋅ ∣ η ) ρ ′ = ρ V ( ⋅ ∣ η ′ ) \rho'=\rho_V(\cdot\mid\eta') ρ ′ = ρ V ( ⋅ ∣ η ′ ) ε = C V , j c ∣ η j − η j ′ ∣ \varepsilon=C^c_{V,j}|\eta_j-\eta'_j| ε = C V , j c ∣ η j − η j ′ ∣ γ V c ( ⋅ ∣ η ′ , v ) \gamma^c_V(\cdot\mid\eta',v) γ V c ( ⋅ ∣ η ′ , v ) γ V c ( ⋅ ∣ η ′ , v ′ ) \gamma^c_V(\cdot\mid\eta',v') γ V c ( ⋅ ∣ η ′ , v ′ ) ∑ w ( v w ′ − v w ) r w \sum_w(v'_w-v_w)r_w ∑ w ( v w ′ − v w ) r w d L ( v , v ′ ) d_L(v,v') d L ( v , v ′ ) L w L_w L w ( r w , y ) (r_w,y) ( r w , y ) K ( r w , y ) = ( 0 , s e w ) K(r_w,y)=(0,s e_w) K ( r w , y ) = ( 0 , s e w ) K K K d d d w w w
∑ e e 2 λ d ( e ) r w ( e ) 2 ≤ ( 2 M 0 s ) 2 = 20160 2 . \sum_e e^{2\lambda d(e)}r_w(e)^2\le(2M_0s)^2=20160^2. e ∑ e 2 λ d ( e ) r w ( e ) 2 ≤ ( 2 M 0 s ) 2 = 2016 0 2 . With z 0 = 2 y + e μ z_0=2y+e_\mu z 0 = 2 y + e μ w = ( y , μ ) w=(y,\mu) w = ( y , μ ) k k k z 0 z_0 z 0 k − 1 k-1 k − 1
L w 2 ≤ 20160 2 ⋅ 4 e 2 λ ( 1 + ∑ k ≥ 1 ( 64 k 3 + 16 k ) e − 2 λ k ) ≤ ( 4 ⋅ 10 14 ) 2 . L_w^2\le20160^2\cdot4e^{2\lambda}\left(1+\sum_{k\ge1}(64k^3+16k)e^{-2\lambda k}\right)\le(4\cdot10^{14})^2. L w 2 ≤ 2016 0 2 ⋅ 4 e 2 λ ( 1 + k ≥ 1 ∑ ( 64 k 3 + 16 k ) e − 2 λk ) ≤ ( 4 ⋅ 1 0 14 ) 2 . Summing over the translates as in Definition 42 gives K c ≤ K c c + K c ρ K_c\le K^c_c+K^\rho_c K c ≤ K c c + K c ρ α / 2 \alpha/2 α /2 α \alpha α K c ρ ≤ α 2 N n K^\rho_c\le\frac{\alpha}{2}N_n K c ρ ≤ 2 α N n
The responses are much smaller than the bound: in floating point, max w L w \max_w L_w max w L w ( m , m , m , m ) (m,m,m,m) ( m , m , m , m ) m = ⌊ n / 4 ⌋ m=\lfloor n/4\rfloor m = ⌊ n /4 ⌋ n = 4 , 6 , 8 n=4,6,8 n = 4 , 6 , 8 D ( n , α ) D(n,\alpha) D ( n , α ) K c c K^c_c K c c d L d_L d L S k S_k S k
Remark 92. Theorem 88 is the weak-coupling half of the constrained route at leading order, and Theorem 45 cannot use it as it stands. Four steps remain, each an open problem.
(a) Composition. D c D_c D c Theorem 45 . Corollary 91 is one step of it, and Proposition 89(iv) gives the gap at every scale; the localisation of S k S_k S k k k k
(b) Beyond the quadratic approximation. For the Wilson measure the block averages are products of group elements along paths, their level sets are curved, and the Bakry–Émery tensor of Sec. VI is negative at large fields; Theorem 85 controls only the small fields, and the large fields have to be split off as in Batabăni’s analysis [7 ] . For the straight paths between block roots the group-valued estimate holds for all fields (Theorem 94 ), but that map does not compose across scales (Proposition 97), and for every compact connected group no continuous covariant map treats the transverse positions of a block alike (Proposition 98). The cut-off average of Theorem 100 is coercive for all fields and agrees with the averages to first order, with constants for one step; on torus-valued fields of small curvature it composes (Theorem 102 ), and for every compact group it composes on fields whose curvature is small at every scale with a small sum (Theorem 106 ); under the Wilson measure the remaining large fields form microscopic clusters (Corollary 109), but the conditional estimate needed near them is open.
(c) Second order. Let b 0 = 11 N / ( 48 π 2 ) b_0=11N/(48\pi^2) b 0 = 11 N / ( 48 π 2 ) SU ( N ) \operatorname{SU}(N) SU ( N ) t = 2 b 0 log 2 t=2b_0\log2 t = 2 b 0 log 2 g m − 2 = g − 2 + t m g_m^{-2}=g^{-2}+tm g m − 2 = g − 2 + t m m m m ∑ m ≤ M g m 2 = t − 1 log M + O ( 1 ) \sum_{m\le M}g_m^2=t^{-1}\log M+O(1) ∑ m ≤ M g m 2 = t − 1 log M + O ( 1 ) ∑ m g m 4 ≤ π 2 / ( 6 t 2 ) \sum_m g_m^4\le\pi^2/(6t^2) ∑ m g m 4 ≤ π 2 / ( 6 t 2 ) [53 ] . A loss 1 + c g m 2 1+cg_m^2 1 + c g m 2 M M M g 2 g^2 g 2 g 4 g^4 g 4
(d) Crossover and existence. The passage from weak coupling to the strong-coupling range of Sec. IV–VIII and the existence of the limit, (E), are untouched.
The renormalisation group needed for (a) and (b) is the ultraviolet half of (E), so the constrained route places both residues on one construction.
Path blocks for the group-valued links
Remark 92(b) asks for estimates on the actual group-valued links. They can be proved for a coarser block map, the straight path between block roots, and this map exhibits the second half of the difficulty: it does not compose across scales. A topological argument shows why no continuous map that treats the transverse positions alike can take its place.
Let G ⊂ U ( r ) G\subset\operatorname{U}(r) G ⊂ U ( r ) δ ( A , B ) = ∥ A − B ∥ H S \delta(A,B)=\|A-B\|_{\mathrm{HS}} δ ( A , B ) = ∥ A − B ∥ HS 2 L 2L 2 L I I I B \mathcal{B} B
V y , μ p ( U ) = U 2 y + e μ , μ , V^p_{y,\mu}(U)=U_{2y+e_\mu,\mu}, V y , μ p ( U ) = U 2 y + e μ , μ , the product along the straight path from 2 y 2y 2 y 2 y + 2 e μ 2y+2e_\mu 2 y + 2 e μ V p = v V^p=v V p = v G 45 B G^{45\mathcal{B}} G 45 B reconstruction R ( v ) R(v) R ( v ) R 2 y + o , μ = v y , μ R_{2y+o,\mu}=v_{y,\mu} R 2 y + o , μ = v y , μ o μ = 1 o_\mu=1 o μ = 1 R 2 y + o , μ = I R_{2y+o,\mu}=I R 2 y + o , μ = I U U U v v v
D v ( U ) 2 = ∑ e δ ( U e , R e ( v ) ) 2 , Q ( U ) = ∑ p δ ( U p , I ) 2 , D_v(U)^2=\sum_e\delta(U_e,R_e(v))^2,\qquad Q(U)=\sum_p\delta(U_p,I)^2, D v ( U ) 2 = e ∑ δ ( U e , R e ( v ) ) 2 , Q ( U ) = p ∑ δ ( U p , I ) 2 , ต่ํา the first sum having zero terms on the tree and path links, and let Q 1 ( v ) = ∑ P δ ( v P , I ) 2 Q_1(v)=\sum_P\delta(v_P,I)^2 Q 1 ( v ) = ∑ P δ ( v P , I ) 2
Lemma 93. For vectors z a z_a z a a ∈ { 0 , 1 } 3 a\in\{0,1\}^3 a ∈ { 0 , 1 } 3 z 0 = 0 z_0=0 z 0 = 0
∑ a ∥ z a ∥ 2 ≤ 3 + 7 2 ∑ { a , a ′ } ∥ z a − z a ′ ∥ 2 ≤ 3 ∑ { a , a ′ } ∥ z a − z a ′ ∥ 2 . \sum_a\lVert z_a\rVert^2\leq\frac{3+\sqrt{7}}{2}\sum_{\{a,a'\}}\lVert z_a-z_{a'}\rVert^2\leq3\sum_{\{a,a'\}}\lVert z_a-z_{a'}\rVert^2. a ∑ ∥ z a ∥ 2 ≤ 2 3 + 7 { a , a ′ } ∑ ∥ z a − z a ′ ∥ 2 ≤ 3 { a , a ′ } ∑ ∥ z a − z a ′ ∥ 2 . the sums on the right over the twelve edges of the cube.
Proof. It suffices to treat real numbers. The right side is ⟨ z , L 0 z ⟩ \langle z,L_0z\rangle ⟨ z , L 0 z ⟩ L 0 L_0 L 0 0 0 0 3 − 7 3-\sqrt{7} 3 − 7 L 0 L_0 L 0 f k f_k f k k = ∣ a ∣ k=|a| k = ∣ a ∣ L 0 L_0 L 0 f 1 ↦ 3 f 1 − 2 f 2 f_1\mapsto3f_1-2f_2 f 1 ↦ 3 f 1 − 2 f 2 f 2 ↦ 3 f 2 − 2 f 1 − f 3 f_2\mapsto3f_2-2f_1-f_3 f 2 ↦ 3 f 2 − 2 f 1 − f 3 f 3 ↦ 3 f 3 − 3 f 2 f_3\mapsto3f_3-3f_2 f 3 ↦ 3 f 3 − 3 f 2 t = 3 − λ t=3-\lambda t = 3 − λ t 3 − 7 t t^3-7t t 3 − 7 t 3 3 3 3 ± 7 3\pm\sqrt{7} 3 ± 7 L 0 L_0 L 0 ( 3 1 1 3 ) \begin{pmatrix}3&1\\1&3\end{pmatrix} ( 3 1 1 3 ) 2 2 2 4 4 4 3 3 3 4 4 4 R 7 \mathbb{R}^7 R 7 1 / ( 3 − 7 ) = ( 3 + 7 ) / 2 < 3 1/(3-\sqrt{7})=(3+\sqrt{7})/2<3 1/ ( 3 − 7 ) = ( 3 + 7 ) /2 < 3
Theorem 94. For every G ⊂ U ( r ) G\subset\mathrm{U}(r) G ⊂ U ( r ) L ≥ 1 L\geq1 L ≥ 1 v v v U U U v v v
D v ( U ) 2 ≤ 224 Q ( U ) . D_v(U)^2\leq224Q(U). D v ( U ) 2 ≤ 224 Q ( U ) . Proof. Let I 0 I_0 I 0 δ ( U e , I ) 2 \delta(U_e,I)^2 δ ( U e , I ) 2 J 0 J_0 J 0 δ ( U e , v y , μ ) 2 \delta(U_e,v_{y,\mu})^2 δ ( U e , v y , μ ) 2 B ( y ) B(y) B ( y ) μ \mu μ D v 2 = I 0 + J 0 D_v^2=I_0+J_0 D v 2 = I 0 + J 0 δ \delta δ δ ( U e , I ) 2 ≤ k ∑ i δ ( U p i , I ) 2 \delta(U_e,I)^2\leq k\sum_i\delta(U_{p_i},I)^2 δ ( U e , I ) 2 ≤ k ∑ i δ ( U p i , I ) 2 k ≤ 3 k\leq3 k ≤ 3 I 0 ≤ 8 Q i n I_0\leq8Q_{\rm in} I 0 ≤ 8 Q in y , μ y,\mu y , μ B ( y ) B(y) B ( y ) a ∈ { 0 , 1 } 3 a\in\{0,1\}^3 a ∈ { 0 , 1 } 3 a = 0 a=0 a = 0 v y , μ v_{y,\mu} v y , μ { a , a ′ } \{a,a'\} { a , a ′ } U a ψ U a ′ − 1 ψ ′ − 1 U_a\psi U_{a'}^{-1}\psi'^{-1} U a ψ U a ′ − 1 ψ ′ − 1 ψ , ψ ′ \psi,\psi' ψ , ψ ′ δ ( U a , U a ′ ) ≤ δ ( U p , I ) + δ ( ψ , I ) + δ ( ψ ′ , I ) \delta(U_a,U_{a'})\leq\delta(U_p,I)+\delta(\psi,I)+\delta(\psi',I) δ ( U a , U a ′ ) ≤ δ ( U p , I ) + δ ( ψ , I ) + δ ( ψ ′ , I ) z a = U a − v y , μ z_a=U_a-v_{y,\mu} z a = U a − v y , μ J 0 ≤ 9 Q f a c e + 27 I 0 J_0\leq9Q_{\rm face}+27I_0 J 0 ≤ 9 Q face + 27 I 0 D v 2 ≤ 28 I 0 + 9 Q f a c e ≤ 224 Q D_v^2\leq28I_0+9Q_{\rm face}\leq224Q D v 2 ≤ 28 I 0 + 9 Q face ≤ 224 Q
The bound holds for all fields, with no small-field condition and no logarithmic chart; Theorem 85 is its linearisation for the averages, with the better constant 52.
Proposition 95. Q 1 ( V p ( U ) ) ≤ 4 Q ( U ) Q_1(V^p(U))\leq4Q(U) Q 1 ( V p ( U )) ≤ 4 Q ( U ) Q ( R ( v ) ) = 4 Q 1 ( v ) Q(R(v))=4Q_1(v) Q ( R ( v )) = 4 Q 1 ( v ) Q ( U ) ≤ 8 Q 1 ( v ) + 48 D v ( U ) 2 Q(U)\leq8Q_1(v)+48D_v(U)^2 Q ( U ) ≤ 8 Q 1 ( v ) + 48 D v ( U ) 2 U U U v v v Q 1 ( v ) = 0 Q_1(v)=0 Q 1 ( v ) = 0 Q ( U ) ≤ 24 D v ( U ) 2 Q(U)\leq24D_v(U)^2 Q ( U ) ≤ 24 D v ( U ) 2
Proof. The holonomy of a coarse plaquette is that of the square of side two at 2 y 2y 2 y δ ( v p , I ) ≤ ∑ p ⊂ P δ ( U p , I ) \delta(v_{p},I)\leq\sum_{p\subset P}\delta(U_p,I) δ ( v p , I ) ≤ ∑ p ⊂ P δ ( U p , I ) R ( v ) R(v) R ( v ) P P P δ ( U p , R p ) ≤ ∑ e ∈ p δ ( U e , R e ) \delta(U_p,R_p)\leq\sum_{e\in p}\delta(U_e,R_e) δ ( U p , R p ) ≤ ∑ e ∈ p δ ( U e , R e ) δ ( U p , I ) 2 ≤ 2 δ ( R p , I ) 2 + 8 ∑ e ∈ p δ ( U e , R e ) 2 \delta(U_p,I)^2\leq2\delta(R_p,I)^2+8\sum_{e\in p}\delta(U_e,R_e)^2 δ ( U p , I ) 2 ≤ 2 δ ( R p , I ) 2 + 8 ∑ e ∈ p δ ( U e , R e ) 2 Q 1 ( v ) = 0 Q_1(v)=0 Q 1 ( v ) = 0 R p R_p R p I I I
Let m = 45 B m=45\mathcal{B} m = 45 B λ Q \lambda Q λ Q λ ≥ 0 \lambda\geq0 λ ≥ 0 S U ( N ) \mathrm{SU}(N) SU ( N ) λ = β / ( 2 N ) \lambda=\beta/(2N) λ = β / ( 2 N )
z G ( t ) = ∫ G e − t δ ( g , I ) 2 d g , Z λ ( v ) = ∫ G m e − λ Q ( U ) d U , z_G(t)=\int_G e^{-t\delta(g,I)^2}\,dg,\qquad Z_\lambda(v)=\int_{G^m}e^{-\lambda Q(U)}\,dU, z G ( t ) = ∫ G e − t δ ( g , I ) 2 d g , Z λ ( v ) = ∫ G m e − λ Q ( U ) d U , with normalised Haar measures, the integral over the fibre of v v v
Theorem 96. For every v v v
e − 8 λ Q 1 ( v ) z G ( 48 λ ) m ≤ Z λ ( v ) ≤ e − λ Q 1 ( v ) / 8 z G ( λ / 448 ) m , e^{-8\lambda Q_1(v)}z_G(48\lambda)^m\leq Z_\lambda(v)\leq e^{-\lambda Q_1(v)/8}z_G(\lambda/448)^m, e − 8 λ Q 1 ( v ) z G ( 48 λ ) m ≤ Z λ ( v ) ≤ e − λ Q 1 ( v ) /8 z G ( λ /448 ) m , and if Q 1 ( v ) = 0 Q_1(v)=0 Q 1 ( v ) = 0 z G ( 24 λ ) m ≤ Z λ ( v ) ≤ z G ( λ / 224 ) m z_G(24\lambda)^m\leq Z_\lambda(v)\leq z_G(\lambda/224)^m z G ( 24 λ ) m ≤ Z λ ( v ) ≤ z G ( λ /224 ) m A λ = z G ( λ / 448 ) / z G ( 24 λ ) A_\lambda=z_G(\lambda/448)/z_G(24\lambda) A λ = z G ( λ /448 ) / z G ( 24 λ ) v v v
E e λ D v 2 / 448 ≤ A λ m , E δ ( U e , I ) 2 ≤ 20160 λ log A λ \mathbb{E}e^{\lambda D_v^2/448}\leq A_\lambda^m,\qquad\mathbb{E}\delta(U_e,I)^2\leq\frac{20160}{\lambda}\log A_\lambda E e λ D v 2 /448 ≤ A λ m , E δ ( U e , I ) 2 ≤ λ 20160 log A λ for each link e e e v ≡ I v\equiv I v ≡ I A λ → 10752 q / 2 A_\lambda\to10752^{q/2} A λ → 1075 2 q /2 λ → ∞ \lambda\to\infty λ → ∞ G G G q ≥ 1 q\geq1 q ≥ 1
Proof. By Theorem 94 and Proposition 95, Q ≥ 1 2 ( D v 2 / 224 + Q 1 / 4 ) Q\geq\frac12(D_v^2/224+Q_1/4) Q ≥ 2 1 ( D v 2 /224 + Q 1 /4 ) Q ≤ 8 Q 1 + 48 D v 2 Q\leq8Q_1+48D_v^2 Q ≤ 8 Q 1 + 48 D v 2 D v 2 D_v^2 D v 2 R e ( v ) − 1 R_e(v)^{-1} R e ( v ) − 1 ∫ e − t D v 2 d U = z G ( t ) m \int e^{-tD_v^2}\,dU=z_G(t)^m ∫ e − t D v 2 d U = z G ( t ) m Q ≤ 24 D v 2 Q\leq24D_v^2 Q ≤ 24 D v 2 − λ Q + λ D v 2 / 448 ≤ − λ D v 2 / 448 -\lambda Q+\lambda D_v^2/448\leq-\lambda D_v^2/448 − λ Q + λ D v 2 /448 ≤ − λ D v 2 /448 E D v 2 ≤ ( 448 m / λ ) log A λ \mathbb{E}D_v^2\leq(448m/\lambda)\log A_\lambda E D v 2 ≤ ( 448 m / λ ) log A λ v ≡ I v\equiv I v ≡ I B \mathcal{B} B 448 ⋅ 45 = 20160 448\cdot45=20160 448 ⋅ 45 = 20160 δ ( e X , I ) 2 = ∥ X ∥ 2 + O ( ∥ X ∥ 4 ) \delta(e^X,I)^2=\lVert X\rVert^2+O(\lVert X\rVert^4) δ ( e X , I ) 2 = ∥ X ∥ 2 + O (∥ X ∥ 4 ) 0 0 0 δ ( g , I ) \delta(g,I) δ ( g , I ) I I I z G ( t ) ∼ c G t − q / 2 z_G(t)\sim c_Gt^{-q/2} z G ( t ) ∼ c G t − q /2 λ / 448 \lambda/448 λ /448 24 λ 24\lambda 24 λ ( 24 ⋅ 448 ) q / 2 (24\cdot448)^{q/2} ( 24 ⋅ 448 ) q /2
So the fluctuations of the free links around a flat path field are of order λ − 1 / 2 \lambda^{-1/2} λ − 1/2 V p k V_p^k V p k k k k s = 2 k s=2^k s = 2 k Q k Q_k Q k s s s
Proposition 97. On the periodic lattice of side 3 s 3s 3 s s = 2 k s=2^k s = 2 k S U ( 2 ) \mathrm{SU}(2) SU ( 2 )
Q ( U ) Q k ( V p , k ( U ) ) = s sin 2 π 2 s ≤ π 2 4 s . \frac{Q(U)}{Q_k(V^{p,k}(U))}=s\sin^2\frac{\pi}{2s}\leq\frac{\pi^2}{4s}. Q k ( V p , k ( U )) Q ( U ) = s sin 2 2 s π ≤ 4 s π 2 . For the averages, by contrast, Q k ( V ( k ) ( φ ) ) ≤ Q ( φ ) Q_k(V^{(k)}(\varphi))\leq Q(\varphi) Q k ( V ( k ) ( φ )) ≤ Q ( φ ) k k k
Proof. Let T = diag ( i π / 2 , − i π / 2 ) T=\operatorname{diag}(i\pi/2,-i\pi/2) T = diag ( iπ /2 , − iπ /2 ) s / 2 s/2 s /2 0 0 0 ( j , 0 , 0 , 0 ) (j,0,0,0) ( j , 0 , 0 , 0 ) j j j j < s j<s j < s e 2 T / s e^{2T/s} e 2 T / s I I I 0 0 0 j j j 0 0 0 s e 0 se_0 s e 0 e T e^T e T I I I e T e^T e T δ 2 = ∥ e T − I ∥ H S 2 = 4 \delta^2=\lVert e^T-I\rVert_{\mathrm{HS}}^2=4 δ 2 = ∥ e T − I ∥ HS 2 = 4 Q k = 24 Q_k=24 Q k = 24 0 0 0 3 s 3s 3 s δ 2 = 8 sin 2 ( π / 2 s ) \delta^2=8\sin^2(\pi/2s) δ 2 = 8 sin 2 ( π /2 s ) Q = 24 s sin 2 ( π / 2 s ) Q=24s\sin^2(\pi/2s) Q = 24 s sin 2 ( π /2 s )
A field concentrated on one thin path carries a coarse curvature of order one at a fine cost of order 1 / s 1/s 1/ s s 3 s^3 s 3 s − 3 s^{-3} s − 3
A block map with both properties cannot treat the transverse positions alike and be continuous, for any compact connected gauge group. Let s = 2 k s=2^k s = 2 k V V V B ( y ) ∪ B ( y + e μ ) B(y)\cup B(y+e_\mu) B ( y ) ∪ B ( y + e μ ) s s s G G G V ( U h ) = h V ( U ) h 1 − 1 V(U^h)=hV(U)h_1^{-1} V ( U h ) = hV ( U ) h 1 − 1 h h h h 1 h_1 h 1 B ( y ) B(y) B ( y ) h 2 h_2 h 2 B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) V ( I ) = I V(I)=I V ( I ) = I T T T s 3 s^3 s 3 μ \mu μ B ( y ) B(y) B ( y ) B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) u ∈ G T u\in G^T u ∈ G T U ( u ) U(u) U ( u ) u t u_t u t t ∈ T t\in T t ∈ T I I I Figure 13 ). Let S ⊂ G S\subset G S ⊂ G ℓ \ell ℓ S S S ( R / 2 π Z ) ℓ (\mathbb{R}/2\pi\mathbb{Z})^\ell ( R /2 π Z ) ℓ Z ℓ \mathbb{Z}^\ell Z ℓ
Figure 13. (a) Two adjacent blocks B ( y ) B(y) B ( y ) B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) s = 4 s=4 s = 4 T T T 1 / s 1/s 1/ s SU ( 2 ) \operatorname{SU}(2) SU ( 2 ) ( R / 2 π Z ) T (\mathbb{R}/2\pi\mathbb{Z})^T ( R /2 π Z ) T W t W_t W t
Proposition 98. Let G ⊂ U ( r ) G\subset U(r) G ⊂ U ( r ) V V V F ( u ) = V ( U ( u ) ) F(u)=V(U(u)) F ( u ) = V ( U ( u )) G T G^T G T F F F S T S^T S T S S S W t W_t W t u t ↦ F ( u ) u_t\mapsto F(u) u t ↦ F ( u ) u t ∈ S u_t\in S u t ∈ S u t ′ = I u_{t'}=I u t ′ = I t ′ ≠ t t'\ne t t ′ = t H 1 ( S ) = Z ℓ H_1(S)=\mathbb{Z}^{\ell} H 1 ( S ) = Z ℓ
∑ t ∈ T W t = I ℓ . \sum_{t\in T}W_t=I_\ell. t ∈ T ∑ W t = I ℓ . Hence some W t ≠ 0 W_t\ne0 W t = 0 u t ↦ V ( U ( u ) ) u_t\mapsto V(U(u)) u t ↦ V ( U ( u )) I I I t t t S S S ∣ T ∣ ≥ 2 |T|\geq2 ∣ T ∣ ≥ 2 T T T F F F G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) ℓ = 1 \ell=1 ℓ = 1 W t W_t W t I I I T T T V = − I V=-I V = − I
Proof. The gauge transformation equal to g g g B ( y ) ∪ B ( y + e μ ) B(y)\cup B(y+e_\mu) B ( y ) ∪ B ( y + e μ ) U ( u ) U(u) U ( u ) U ( g u − 1 ) U(gu^{-1}) U ( g u − 1 ) I I I I I I g g g B ( y ) B(y) B ( y ) I I I B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) U ( g u ) U(gu) U ( g u ) F ( g u g − 1 ) = g F ( u ) g − 1 F(gug^{-1})=gF(u)g^{-1} F ( g u g − 1 ) = g F ( u ) g − 1 F ( g u ) = g F ( u ) F(gu)=gF(u) F ( g u ) = g F ( u ) F ( I ) = I F(I)=I F ( I ) = I F ( g , … , g ) = g F(g,\ldots,g)=g F ( g , … , g ) = g u t u_t u t S S S g ∈ S g\in S g ∈ S F ( u ) F(u) F ( u ) S S S [42 ] , so F ( u ) ∈ S F(u)\in S F ( u ) ∈ S F F F f : S T → S f:S^T\to S f : S T → S S T S^T S T ⨁ t ∈ T Z ℓ \bigoplus_{t\in T}\mathbb{Z}^{\ell} ⨁ t ∈ T Z ℓ t t t t t t f f f ( x t ) t ↦ ∑ t W t x t (x_t)_t\mapsto\sum_t W_tx_t ( x t ) t ↦ ∑ t W t x t S → S T S\to S^T S → S T x ↦ ( x , … , x ) x\mapsto(x,\ldots,x) x ↦ ( x , … , x ) f f f S S S F ( g , … , g ) = g F(g,\ldots,g)=g F ( g , … , g ) = g ∑ t W t = I ℓ \sum_tW_t=I_\ell ∑ t W t = I ℓ W t W_t W t F F F W t W_t W t W W W ∣ T ∣ W = I ℓ |T|W=I_\ell ∣ T ∣ W = I ℓ ∣ T ∣ ≥ 2 |T|\geq2 ∣ T ∣ ≥ 2 S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) S S S − I -I − I
For the path map, W t W_t W t G T G^T G T S U ( N ) \mathrm{SU}(N) SU ( N ) [5 , 7 ] . A block map with both properties for all fields is therefore discontinuous or, like the path map, not invariant under the transverse symmetries of the block. Theorem 100 below gives a continuous one whose departure from the symmetry is confined to large fields.19
The proposition leaves room for a map that treats the positions alike to first order and breaks the symmetry only where the field is large. For a coarse link ( y , μ ) (y,\mu) ( y , μ ) x ∈ B ( y ) x\in B(y) x ∈ B ( y )
P x = U x , μ U x + e μ , μ , P_x=U_{x,\mu}U_{x+e_\mu,\mu}, P x = U x , μ U x + e μ , μ , the product along the straight path of length two from x x x x + 2 e μ x+2e_\mu x + 2 e μ 2 y 2y 2 y 2 y + 2 e μ 2y+2e_\mu 2 y + 2 e μ x x x x + 2 e μ x+2e_\mu x + 2 e μ P 2 y = V y , μ p P_{2y}=V^p_{y,\mu} P 2 y = V y , μ p g \mathfrak{g} g G G G ∥ X ∥ \lVert X\rVert ∥ X ∥ log \log log − 1 -1 − 1
Lemma 99. If g g g δ ( g , I ) ≤ 1 \delta(g,I)\leq1 δ ( g , I ) ≤ 1 ∥ log g ∥ ≤ π 3 δ ( g , I ) \lVert\log g\rVert\leq\frac{\pi}{3}\delta(g,I) ∥ log g ∥ ≤ 3 π δ ( g , I ) X X X δ ( e X , I ) ≤ ∥ X ∥ \delta(e^X,I)\leq\lVert X\rVert δ ( e X , I ) ≤ ∥ X ∥
Proof. Let e i θ j e^{i\theta_j} e i θ j ∣ θ j ∣ ≤ π |\theta_j|\leq\pi ∣ θ j ∣ ≤ π g g g δ ( g , I ) 2 = ∑ j 4 sin 2 ( θ j / 2 ) \delta(g,I)^2=\sum_j4\sin^2(\theta_j/2) δ ( g , I ) 2 = ∑ j 4 sin 2 ( θ j /2 ) ∥ log g ∥ 2 = ∑ j θ j 2 \lVert\log g\rVert^2=\sum_j\theta_j^2 ∥ log g ∥ 2 = ∑ j θ j 2 1 1 1 ∣ θ j ∣ ≤ π / 3 |\theta_j|\leq\pi/3 ∣ θ j ∣ ≤ π /3 θ / ( 2 sin ( θ / 2 ) ) \theta/(2\sin(\theta/2)) θ / ( 2 sin ( θ /2 )) ( 0 , π ) (0,\pi) ( 0 , π ) π / 3 \pi/3 π /3 θ = π / 3 \theta=\pi/3 θ = π /3 ∣ e i θ − 1 ∣ ≤ ∣ θ ∣ |e^{i\theta}-1|\leq|\theta| ∣ e i θ − 1∣ ≤ ∣ θ ∣ X X X
Let ρ G ∈ ( 0 , 1 ] \rho_G\in(0,1] ρ G ∈ ( 0 , 1 ] log g ∈ g \log g\in\mathfrak{g} log g ∈ g g ∈ G g\in G g ∈ G δ ( g , I ) ≤ ρ G \delta(g,I)\leq\rho_G δ ( g , I ) ≤ ρ G 20 ϕ ( t ) = min { 1 , max { 0 , 2 − 2 t } } \phi(t)=\min\{1,\max\{0,2-2t\}\} ϕ ( t ) = min { 1 , max { 0 , 2 − 2 t }} 1 1 1 [ 0 , 1 2 ] [0,\frac{1}{2}] [ 0 , 2 1 ] 0 0 0 [ 1 , ∞ ) [1,\infty) [ 1 , ∞ )
χ y , μ ( U ) = ϕ ( ρ G − 1 max x ∈ B ( y ) δ ( P x , P 2 y ) ) , \chi_{y,\mu}(U)=\phi\left(\rho_G^{-1}\max_{x\in B(y)}\delta(P_x,P_{2y})\right), χ y , μ ( U ) = ϕ ( ρ G − 1 x ∈ B ( y ) max δ ( P x , P 2 y ) ) , and define the cut-off average
V y , μ ♯ ( U ) = P 2 y exp ( χ y , μ ( U ) 16 ∑ x ∈ B ( y ) log ( P 2 y − 1 P x ) ) . (20) V^\sharp_{y,\mu}(U)=P_{2y}\exp\left(\frac{\chi_{y,\mu}(U)}{16}\sum_{x\in B(y)}\log(P_{2y}^{-1}P_x)\right). \tag*{(20)} V y , μ ♯ ( U ) = P 2 y exp 16 χ y , μ ( U ) x ∈ B ( y ) ∑ log ( P 2 y − 1 P x ) . (20) the exponential factor being I I I ({xref}Fig. 14`). When every path is close to the straight one it is the average of the sixteen paths taken in the chart at P 2 y P_{2y} P 2 y
Theorem 100. For every compact G ⊂ U ( r ) G\subset\operatorname{U}(r) G ⊂ U ( r ) c = π / 3 c=\pi/3 c = π /3
(i) If δ ( P x , P 2 y ) ≤ ρ G / 2 \delta(P_x,P_{2y})\leq\rho_G/2 δ ( P x , P 2 y ) ≤ ρ G /2 x ∈ B ( y ) x\in B(y) x ∈ B ( y )
V y , μ ♯ ( U ) = P 2 y exp ( 1 16 ∑ x log ( P 2 y − 1 P x ) ) . V^\sharp_{y,\mu}(U)=P_{2y}\exp\left(\frac{1}{16}\sum_x\log(P_{2y}^{-1}P_x)\right). V y , μ ♯ ( U ) = P 2 y exp ( 16 1 x ∑ log ( P 2 y − 1 P x ) ) . For φ \varphi φ g \mathfrak{g} g
V y , μ ♯ ( e t φ ) = I + t V y , μ ( φ ) + O ( t 2 ) , V^\sharp_{y,\mu}(e^{t\varphi})=I+tV_{y,\mu}(\varphi)+O(t^2), V y , μ ♯ ( e tφ ) = I + t V y , μ ( φ ) + O ( t 2 ) , with V y , μ V_{y,\mu} V y , μ
(ii) For every field U U U
∑ y , μ ∑ x ∈ B ( y ) δ ( P x , P 2 y ) 2 ≤ 896 Q ( U ) , \sum_{y,\mu}\sum_{x\in B(y)}\delta(P_x,P_{2y})^2\leq896Q(U), y , μ ∑ x ∈ B ( y ) ∑ δ ( P x , P 2 y ) 2 ≤ 896 Q ( U ) , ∑ y , μ δ ( V y , μ ♯ ( U ) , V y , μ ρ ( U ) ) 2 ≤ 56 c 2 Q ( U ) , \sum_{y,\mu}\delta(V^\sharp_{y,\mu}(U),V^\rho_{y,\mu}(U))^2\leq56c^2Q(U), y , μ ∑ δ ( V y , μ ♯ ( U ) , V y , μ ρ ( U ) ) 2 ≤ 56 c 2 Q ( U ) , ∑ e δ ( U e , Re ( V ♯ ( U ) ) ) 2 ≤ ( 448 + 896 c 2 ) Q ( U ) , \sum_e\delta(U_e,\operatorname{Re}(V^\sharp(U)))^2\leq(448+896c^2)Q(U), e ∑ δ ( U e , Re ( V ♯ ( U )) ) 2 ≤ ( 448 + 896 c 2 ) Q ( U ) , Q 1 ( V ♯ ( U ) ) ≤ ( 8 + 2688 c 2 ) Q ( U ) , Q_1(V^\sharp(U))\leq(8+2688c^2)Q(U), Q 1 ( V ♯ ( U )) ≤ ( 8 + 2688 c 2 ) Q ( U ) , and 448 + 896 c 2 < 1431 448+896c^2<1431 448 + 896 c 2 < 1431 8 + 2688 c 2 < 2956 8+2688c^2<2956 8 + 2688 c 2 < 2956
(iii) In the setting of Proposition 98 with s = 2 s=2 s = 2 W t W_t W t is the identity at the position of the straight path and 0 0 0 at the seven others.
(iv) If U U U equals I I I except on one link from B ( y ) B(y) B ( y ) to B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) where it equals g g g with δ ( g , I ) ≤ ρ G / 2 \delta(g,I)\leq\rho_G/2 δ ( g , I ) ≤ ρ G /2 then V y , μ ♯ ( U ) = exp ( 1 8 log g ) V^\sharp_{y,\mu}(U)=\exp(\frac{1}{8}\log g) V y , μ ♯ ( U ) = exp ( 8 1 log g ) at each of the eight transverse positions of the link; the path map gives g g g at one position and I I I at the seven others.
Proof. Write P = P 2 y P=P_{2y} P = P 2 y v x = P − 1 P x v_x=P^{-1}P_x v x = P − 1 P x m = max x δ ( v x , I ) = max x δ ( P x , P ) m=\max_x\delta(v_x,I)=\max_x\delta(P_x,P) m = max x δ ( v x , I ) = max x δ ( P x , P ) m < ρ G m<\rho_G m < ρ G log v x \log v_x log v x g \mathfrak{g} g U U U v x v_x v x − 1 -1 − 1 g \mathfrak{g} g V ♯ ∈ G V^\sharp\in G V ♯ ∈ G m ≥ ρ G m\geq\rho_G m ≥ ρ G V ♯ = P V^\sharp=P V ♯ = P m m m ρ G \rho_G ρ G χ c ρ G → 0 \chi c\rho_G\to0 χc ρ G → 0 V ♯ V^\sharp V ♯ h h h h 1 h_1 h 1 B ( y ) B(y) B ( y ) h 2 h_2 h 2 B ( y + e μ ) B(y+e_\mu) B ( y + e μ ) x + e μ x+e_\mu x + e μ P x ↦ h 1 P x h 2 − 1 P_x\mapsto h_1P_xh_2^{-1} P x ↦ h 1 P x h 2 − 1 v x ↦ h 2 v x h 2 − 1 v_x\mapsto h_2v_xh_2^{-1} v x ↦ h 2 v x h 2 − 1 δ \delta δ log \log log χ \chi χ V ♯ ↦ h 1 V ♯ h 2 − 1 V^\sharp\mapsto h_1V^\sharp h_2^{-1} V ♯ ↦ h 1 V ♯ h 2 − 1 U = I U=I U = I v x v_x v x I I I V ♯ ( I ) = I V^\sharp(I)=I V ♯ ( I ) = I χ = 1 \chi=1 χ = 1 U = e t φ U=e^{t\varphi} U = e tφ φ = 0 \varphi=0 φ = 0 P = I + t φ 2 y + e μ , μ + O ( t 2 ) P=I+t\varphi_{2y+e_\mu,\mu}+O(t^2) P = I + t φ 2 y + e μ , μ + O ( t 2 ) log v x = t ( φ x , μ + φ x + e μ , μ − φ 2 y + e μ , μ ) + O ( t 2 ) \log v_x=t(\varphi_{x,\mu}+\varphi_{x+e_\mu,\mu}-\varphi_{2y+e_\mu,\mu})+O(t^2) log v x = t ( φ x , μ + φ x + e μ , μ − φ 2 y + e μ , μ ) + O ( t 2 ) t t t x x x t ( V y , μ ( φ ) − φ 2 y + e μ , μ ) + O ( t 2 ) t(V_{y,\mu}(\varphi)-\varphi_{2y+e_\mu,\mu})+O(t^2) t ( V y , μ ( φ ) − φ 2 y + e μ , μ ) + O ( t 2 ) φ 2 y , μ = 0 \varphi_{2y,\mu}=0 φ 2 y , μ = 0 P P P
Figure 14. (a) A block B ( y ) B(y) B ( y ) μ \mu μ I I I P x P_x P x B ( y ) B(y) B ( y ) 2 y 2y 2 y 2 y + 2 e μ 2y+2e_\mu 2 y + 2 e μ P 2 y P_{2y} P 2 y P 2 y P_{2y} P 2 y e θ X e^{\theta X} e θ X X = diag ( i , − i ) X=\operatorname{diag}(i,-i) X = diag ( i , − i ) V ♯ V^\sharp V ♯ ρ G = 1 \rho_G=1 ρ G = 1 θ / 8 \theta/8 θ /8 χ = 1 \chi=1 χ = 1 θ \theta θ 0 0 0
(ii) In the tree gauge P = v y , μ : = V y , μ P ( U ) P=v_{y,\mu}:=V^P_{y,\mu}(U) P = v y , μ := V y , μ P ( U ) o μ = 0 o_\mu=0 o μ = 0 U x + e μ , μ U_{x+e_\mu,\mu} U x + e μ , μ B ( y ) B(y) B ( y ) x x x δ \delta δ δ ( P x , P ) ≤ δ ( U x , μ , I ) + δ ( U x + e μ , μ , v y , μ ) \delta(P_x,P)\le\delta(U_{x,\mu},I)+\delta(U_{x+e_\mu,\mu},v_{y,\mu}) δ ( P x , P ) ≤ δ ( U x , μ , I ) + δ ( U x + e μ , μ , v y , μ ) o μ = 1 o_\mu=1 o μ = 1 μ \mu μ x x x μ \mu μ o μ = 0 o_\mu=0 o μ = 0 ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (a+b)^2\le2a^2+2b^2 ( a + b ) 2 ≤ 2 a 2 + 2 b 2 Theorem 94 , the sum is at most 2 ( 2 I 0 + 2 J 0 ) = 4 D v 2 ≤ 896 Q 2(2I_0+2J_0)=4D_v^2\le896Q 2 ( 2 I 0 + 2 J 0 ) = 4 D v 2 ≤ 896 Q χ > 0 \chi>0 χ > 0
δ ( V ♯ , P ) ≤ ∥ χ 16 ∑ x log v x ∥ ≤ c 16 ∑ x δ ( v x , I ) ≤ c 4 ( ∑ x δ ( P x , P ) 2 ) 1 / 2 . \begin{aligned}
\delta(V^\sharp,P)\le\left\lVert\frac{\chi}{16}\sum_x\log v_x\right\rVert\le\frac{c}{16}\sum_x\delta(v_x,I) \\
&\le\frac{c}{4}\left(\sum_x\delta(P_x,P)^2\right)^{1/2}.
\end{aligned} δ ( V ♯ , P ) ≤ 16 χ x ∑ log v x ≤ 16 c x ∑ δ ( v x , I ) ≤ 4 c ( x ∑ δ ( P x , P ) 2 ) 1/2 . and the second bound follows from the first, 896 c 2 / 16 = 56 c 2 896c^2/16=56c^2 896 c 2 /16 = 56 c 2 R ( V ♯ ) R(V^\sharp) R ( V ♯ ) R ( v ) R(v) R ( v ) o μ = 0 o_\mu=0 o μ = 0 o μ = 1 o_\mu=1 o μ = 1 δ ( U e , V ♯ ) 2 ≤ 2 δ ( U e , v ) 2 + 2 δ ( V ♯ , v ) 2 \delta(U_e,V^\sharp)^2\le2\delta(U_e,v)^2+2\delta(V^\sharp,v)^2 δ ( U e , V ♯ ) 2 ≤ 2 δ ( U e , v ) 2 + 2 δ ( V ♯ , v ) 2 2 D v 2 + 16 ⋅ 56 c 2 Q ≤ ( 448 + 896 c 2 ) Q 2D_v^2+16\cdot56c^2Q\le(448+896c^2)Q 2 D v 2 + 16 ⋅ 56 c 2 Q ≤ ( 448 + 896 c 2 ) Q V ♯ = v M V^\sharp=vM V ♯ = v M M M M δ ( ⋅ , I ) \delta(\mathord\cdot,I) δ ( ⋅ , I ) δ ( M , I ) \delta(M,I) δ ( M , I ) δ ( V P ♯ , I ) 2 ≤ 2 δ ( v P , I ) 2 + 8 ∑ ℓ ∈ P δ ( M ℓ , I ) 2 \delta(V^\sharp_P,I)^2\le2\delta(v_P,I)^2+8\sum_{\ell\in P}\delta(M_\ell,I)^2 δ ( V P ♯ , I ) 2 ≤ 2 δ ( v P , I ) 2 + 8 ∑ ℓ ∈ P δ ( M ℓ , I ) 2 Q 1 ( V ♯ ) ≤ 2 Q 1 ( v ) + 48 ⋅ 56 c 2 Q Q_1(V^\sharp)\le2Q_1(v)+48\cdot56c^2Q Q 1 ( V ♯ ) ≤ 2 Q 1 ( v ) + 48 ⋅ 56 c 2 Q Q 1 ( v ) ≤ 4 Q Q_1(v)\le4Q Q 1 ( v ) ≤ 4 Q c 2 < 1.0967 c^2<1.0967 c 2 < 1.0967
(iii) For s = 2 s=2 s = 2 T T T B ( y ) B(y) B ( y ) I I I P x = u a P_x=u_a P x = u a a a a x x x a a a u t u_t u t u t ′ = I u_{t'}=I u t ′ = I t ′ ≠ t t'\ne t t ′ = t P = I P=I P = I v x v_x v x u t u_t u t I I I F ( u ) = exp ( 1 8 χ log u t ) F(u)=\exp(\frac{1}{8}\chi\log u_t) F ( u ) = exp ( 8 1 χ log u t ) χ = ϕ ( δ ( u t , I ) / ρ G ) \chi=\phi(\delta(u_t,I)/\rho_G) χ = ϕ ( δ ( u t , I ) / ρ G ) S S S u t ↦ exp ( 1 8 χ log u t ) u_t\mapsto\exp(\frac{1}{8}\chi\log u_t) u t ↦ exp ( 8 1 χ log u t ) 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 S S S log u t \log u_t log u t S S S g \mathfrak g g F F F I I I W t = 0 W_t=0 W t = 0 W t W_t W t
(iv) At the straight path, P x = g P_x=g P x = g 0 0 0 P x = I P_x=I P x = I V ♯ = g exp ( − 14 16 log g ) = exp ( 1 8 log g ) V^\sharp=g\exp(-\frac{14}{16}\log g)=\exp(\frac18\log g) V ♯ = g exp ( − 16 14 log g ) = exp ( 8 1 log g ) P = I P=I P = I P x P_x P x g g g V ♯ = exp ( 2 16 log g ) V^\sharp=\exp(\frac{2}{16}\log g) V ♯ = exp ( 16 2 log g )
The small-field part of Eq. (20) is Balaban’s averaging operation [6 ] ; the cut-off extends it to all fields with the bounds of (ii). The theorem shows how far Proposition 98 reaches. The cut-off average is continuous, covariant and coercive for every field, and it agrees with the block averages to first order. Its slice windings are those of the path map, as the proposition demands of any such map, but the departure from the transverse symmetry happens only where some path differs from the straight one by more than ρ G / 2 \rho_G/2 ρ G /2 1 8 \frac18 8 1
What the theorem does not give is composition. The constants of (ii) are those of one step, and iterating them gives a factor that grows exponentially in the number of steps; on its large-field branch the map is the path map, for which Proposition 97 rules out any bound uniform in that number. Whether the k k k Q k ( V ♯ , k ( U ) ) ≤ C Q ( U ) Q_k(V^{\sharp,k}(U))\le CQ(U) Q k ( V ♯ , k ( U )) ≤ C Q ( U ) C C C k k k C = 1.001 C=1.001 C = 1.001 C = e 12060 σ C=e^{12060\sigma} C = e 12060 σ σ \sigma σ
Composition across scales
This subsection proves that the cut-off average composes across all scales on fields with values in a maximal torus whose curvature stays small, which includes the thin paths that defeat the path map, and then that it composes for every compact group on fields whose curvature is small, with a summable loss over the scales.
Let S S S G ⊂ U ( r ) G\subset U(r) G ⊂ U ( r ) s \mathfrak{s} s s \mathfrak{s} s S S S Λ \Lambda Λ 2 π i Z 2\pi i\mathbb{Z} 2 π i Z Λ \Lambda Λ 2 π 2\pi 2 π S S S δ ( U p , I ) ≤ ρ G \delta(U_p,I)\leq\rho_G δ ( U p , I ) ≤ ρ G Θ p = log U p \Theta_p=\log U_p Θ p = log U p g \mathfrak{g} g S S S s \mathfrak{s} s x x x { μ , ν } \{\mu,\nu\} { μ , ν } □ x \square_x □ x x x x F x = ∑ p ⊂ □ x Θ p F_x=\sum_{p\subset\square_x}\Theta_p F x = ∑ p ⊂ □ x Θ p
η G = ρ G 12 . \eta_G=\frac{\rho_G}{12}. η G = 12 ρ G . The k k k S S S V ♯ V^\sharp V ♯ U P ( j ) U_P^{(j)} U P ( j ) j j j U ( j ) U^{(j)} U ( j ) Θ P ( j ) = log U P ( j ) \Theta_P^{(j)}=\log U_P^{(j)} Θ P ( j ) = log U P ( j ) δ ( U P ( j ) , I ) ≤ ρ G \delta(U_P^{(j)},I)\leq\rho_G δ ( U P ( j ) , I ) ≤ ρ G
Lemma 101. Let U U U S S S ∥ Θ p ∥ ≤ η G \|\Theta_p\|\leq\eta_G ∥ Θ p ∥ ≤ η G χ y , μ ( U ) = 1 \chi_{y,\mu}(U)=1 χ y , μ ( U ) = 1 P P P y y y { μ , ν } \{\mu,\nu\} { μ , ν }
V ♯ ( U ) P = exp F ‾ P , F ‾ P = 1 16 ∑ o ∈ { 0 , 1 } 4 F 2 y + o , V^\sharp(U)_P=\exp\overline{F}_P,\qquad\overline{F}_P=\frac{1}{16}\sum_{o\in\{0,1\}^4}F_{2y+o}, V ♯ ( U ) P = exp F P , F P = 16 1 o ∈ { 0 , 1 } 4 ∑ F 2 y + o , with ∥ F ‾ P ∥ ≤ 4 max p ∥ Θ p ∥ \|\overline{F}_P\|\leq4\max_p\|\Theta_p\| ∥ F P ∥ ≤ 4 max p ∥ Θ p ∥ log V ♯ ( U ) P = F ‾ P \log V^\sharp(U)_P=\overline{F}_P log V ♯ ( U ) P = F P
Proof. Let η = max p ∥ Θ p ∥ \eta=\max_p\|\Theta_p\| η = max p ∥ Θ p ∥ ( y , μ ) (y,\mu) ( y , μ ) x = 2 y + o x=2y+o x = 2 y + o τ \tau τ 2 y 2y 2 y x x x o o o μ \mu μ I I I P 2 y − 1 P x P_{2y}^{-1}P_x P 2 y − 1 P x τ \tau τ x x x x + 2 e μ x+2e_\mu x + 2 e μ τ + 2 e μ \tau+2e_\mu τ + 2 e μ 2 y + 2 e μ 2y+2e_\mu 2 y + 2 e μ 2 y 2y 2 y τ \tau τ μ \mu μ ± 1 \pm1 ± 1 P 2 y − 1 P x = exp d x P_{2y}^{-1}P_x=\exp d_x P 2 y − 1 P x = exp d x d x d_x d x Θ p \Theta_p Θ p ∥ d x ∥ ≤ 6 η ≤ ρ G / 2 \|d_x\|\leq6\eta\leq\rho_G/2 ∥ d x ∥ ≤ 6 η ≤ ρ G /2 δ ( P x , P 2 y ) ≤ ρ G / 2 \delta(P_x,P_{2y})\leq\rho_G/2 δ ( P x , P 2 y ) ≤ ρ G /2 χ = 1 \chi=1 χ = 1 d x d_x d x 1 2 \frac12 2 1 log ( P 2 y − 1 P x ) = d x \log(P_{2y}^{-1}P_x)=d_x log ( P 2 y − 1 P x ) = d x
Hence V y , μ ♯ = exp ( X y , μ + d ˉ y , μ ) V^\sharp_{y,\mu}=\exp(X_{y,\mu}+\bar d_{y,\mu}) V y , μ ♯ = exp ( X y , μ + d ˉ y , μ ) X y , μ ∈ s X_{y,\mu}\in\mathfrak{s} X y , μ ∈ s P 2 y P_{2y} P 2 y d ˉ y , μ \bar d_{y,\mu} d ˉ y , μ d x d_x d x
Let the sides of P P P ℓ \ell ℓ ϵ ℓ \epsilon_\ell ϵ ℓ d o , ℓ d_{o,\ell} d o , ℓ d x d_x d x ℓ \ell ℓ o o o o o o o o o □ 2 y + o \square_{2y+o} □ 2 y + o
Y o = ∑ ℓ ϵ ℓ ( X ℓ + d o , ℓ ) − F 2 y + o ∈ Λ . Y_o=\sum_\ell\epsilon_\ell(X_\ell+d_{o,\ell})-F_{2y+o}\in\Lambda. Y o = ℓ ∑ ϵ ℓ ( X ℓ + d o , ℓ ) − F 2 y + o ∈ Λ. Since d o , ℓ = 0 d_{o,\ell}=0 d o , ℓ = 0 ∥ Y o − Y 0 ∥ ≤ 4 ⋅ 6 η + 8 η = 32 η ≤ 8 3 < 2 π \|Y_o-Y_0\|\leq4\cdot6\eta+8\eta=32\eta\leq\frac83<2\pi ∥ Y o − Y 0 ∥ ≤ 4 ⋅ 6 η + 8 η = 32 η ≤ 3 8 < 2 π Y o = Y 0 Y_o=Y_0 Y o = Y 0 o o o o o o ∑ ℓ ϵ ℓ ( X ℓ + d ˉ ℓ ) = F ˉ P + Y 0 \sum_\ell\epsilon_\ell(X_\ell+\bar d_\ell)=\bar F_P+Y_0 ∑ ℓ ϵ ℓ ( X ℓ + d ˉ ℓ ) = F ˉ P + Y 0 exp Y 0 = I \exp Y_0=I exp Y 0 = I ∥ F ˉ P ∥ ≤ max o ∥ F 2 y + o ∥ ≤ 4 η < π / 3 \|\bar F_P\|\leq\max_o\|F_{2y+o}\|\leq4\eta<\pi/3 ∥ F ˉ P ∥ ≤ max o ∥ F 2 y + o ∥ ≤ 4 η < π /3 F ˉ P \bar F_P F ˉ P □ \square □
Theorem 102. Let U U U S S S 2 k 2^k 2 k δ ( U p , I ) ≤ ρ G \delta(U_p,I)\leq\rho_G δ ( U p , I ) ≤ ρ G η 0 = max p ∥ Θ p ∥ \eta_0=\max_p\|\Theta_p\| η 0 = max p ∥ Θ p ∥ A = ∑ p ∥ Θ p ∥ A=\sum_p\|\Theta_p\| A = ∑ p ∥ Θ p ∥
max P ∥ Θ P ( j ) ∥ ≤ η G for 0 ≤ j < k . (21) \max_P\|\Theta_P^{(j)}\|\leq\eta_G\qquad\text{for }0\leq j<k. \tag*{(21)} P max ∥ Θ P ( j ) ∥ ≤ η G for 0 ≤ j < k . (21) which holds in particular if η 0 A ≤ η G 2 \eta_0A\leq\eta_G^2 η 0 A ≤ η G 2 Θ ( j + 1 ) = F ‾ ( Θ ( j ) ) \Theta^{(j+1)}=\overline F(\Theta^{(j)}) Θ ( j + 1 ) = F ( Θ ( j ) ) j < k j<k j < k
∑ P ∥ Θ P ( j + 1 ) ∥ 2 ≤ ∑ P ∥ Θ P ( j ) ∥ 2 , ∑ P ∥ Θ P ( j + 1 ) ∥ ≤ 1 4 ∑ P ∥ Θ P ( j ) ∥ , \sum_P\|\Theta_P^{(j+1)}\|^2\leq\sum_P\|\Theta_P^{(j)}\|^2,\qquad
\sum_P\|\Theta_P^{(j+1)}\|\leq\frac14\sum_P\|\Theta_P^{(j)}\|, P ∑ ∥ Θ P ( j + 1 ) ∥ 2 ≤ P ∑ ∥ Θ P ( j ) ∥ 2 , P ∑ ∥ Θ P ( j + 1 ) ∥ ≤ 4 1 P ∑ ∥ Θ P ( j ) ∥ , max P ∥ Θ P ( j + 1 ) ∥ ≤ 4 max P ∥ Θ P ( j ) ∥ . \max_P\|\Theta_P^{(j+1)}\|\leq4\max_P\|\Theta_P^{(j)}\|. P max ∥ Θ P ( j + 1 ) ∥ ≤ 4 P max ∥ Θ P ( j ) ∥. Consequently
Q k ( V ♯ , k ( U ) ) ≤ ∑ p ∥ Θ p ∥ 2 ≤ 1.001 Q ( U ) , Q_k(V^{\sharp,k}(U))\leq\sum_p\|\Theta_p\|^2\leq1.001Q(U), Q k ( V ♯ , k ( U )) ≤ p ∑ ∥ Θ p ∥ 2 ≤ 1.001 Q ( U ) , Q k ( V ♯ , k ( U ) ) ≤ 16 − k A 2 . Q_k(V^{\sharp,k}(U))\leq16^{-k}A^2. Q k ( V ♯ , k ( U )) ≤ 1 6 − k A 2 . Proof. The first assertions follow from Lemma 101 by induction, the S S S P P P o o o □ x \square_x □ x x = 2 y + o x=2y+o x = 2 y + o
∥ F ˉ P ∥ 2 ≤ 1 16 ∑ o ∥ F 2 y + o ∥ 2 \|\bar F_P\|^2\leq\frac1{16}\sum_{o}\|F_{2y+o}\|^2 ∥ F ˉ P ∥ 2 ≤ 16 1 o ∑ ∥ F 2 y + o ∥ 2 and ∥ F x ∥ 2 ≤ 4 ∑ p ⊂ □ x ∥ Θ p ∥ 2 \|F_x\|^2\leq4\sum_{p\subset\square_x}\|\Theta_p\|^2 ∥ F x ∥ 2 ≤ 4 ∑ p ⊂ □ x ∥ Θ p ∥ 2 ∑ P ∥ F ˉ P ∥ 2 ≤ 1 16 ⋅ 4 ⋅ 4 ∑ p ∥ Θ p ∥ 2 \sum_P\|\bar F_P\|^2\leq\frac1{16}\cdot4\cdot4\sum_p\|\Theta_p\|^2 ∑ P ∥ F ˉ P ∥ 2 ≤ 16 1 ⋅ 4 ⋅ 4 ∑ p ∥ Θ p ∥ 2 1 4 \frac14 4 1 ∥ F x ∥ ≤ 4 η \|F_x\|\leq4\eta ∥ F x ∥ ≤ 4 η
For the sufficient condition, let η j \eta_j η j A j A_j A j ∥ Θ P ( j ) ∥ \|\Theta_P^{(j)}\| ∥ Θ P ( j ) ∥ j − 1 j-1 j − 1 η j ≤ 4 j η 0 \eta_j\leq4^j\eta_0 η j ≤ 4 j η 0 η j ≤ 4 − j A \eta_j\leq4^{-j}A η j ≤ 4 − j A η 0 A ≤ η G 2 \eta_0A\leq\eta_G^2 η 0 A ≤ η G 2 η G \eta_G η G j j j
Finally δ ( exp Θ , I ) ≤ ∥ Θ ∥ \delta(\exp\Theta,I)\leq\|\Theta\| δ ( exp Θ , I ) ≤ ∥Θ∥ Q k ( V ♯ , k ( U ) ) ≤ ∑ P ∥ Θ P ( k ) ∥ 2 Q_k(V^{\sharp,k}(U))\leq\sum_P\|\Theta_P^{(k)}\|^2 Q k ( V ♯ , k ( U )) ≤ ∑ P ∥ Θ P ( k ) ∥ 2 ∑ p ∥ Θ p ∥ 2 \sum_p\|\Theta_p\|^2 ∑ p ∥ Θ p ∥ 2 16 − k A 2 16^{-k}A^2 1 6 − k A 2 Θ p \Theta_p Θ p i θ i\theta i θ ∣ θ ∣ ≤ η 0 ≤ η G ≤ 1 12 |\theta|\leq\eta_0\leq\eta_G\leq\frac1{12} ∣ θ ∣ ≤ η 0 ≤ η G ≤ 12 1 η 0 2 ≤ η 0 A \eta_0^2\leq\eta_0A η 0 2 ≤ η 0 A η 0 A ≤ η G 2 \eta_0A\leq\eta_G^2 η 0 A ≤ η G 2 η G \eta_G η G j = 0 j=0 j = 0 θ 2 ≤ 1.001 ⋅ 4 sin 2 ( θ / 2 ) \theta^2\leq1.001\cdot4\sin^2(\theta/2) θ 2 ≤ 1.001 ⋅ 4 sin 2 ( θ /2 ) ∑ p ∥ Θ p ∥ 2 ≤ 1.001 Q ( U ) \sum_p\|\Theta_p\|^2\leq1.001Q(U) ∑ p ∥ Θ p ∥ 2 ≤ 1.001 Q ( U ) □ \square □
Corollary 103. For G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) s = 2 k ≥ 2 14 s=2^k\geq2^{14} s = 2 k ≥ 2 14
Q k ( V ♯ , k ( U ) ) ≤ 18 π 2 s 2 ≤ 4 π 2 Q ( U ) , Q_k(V^{\sharp,k}(U))\leq\frac{18\pi^2}{s^2}\leq\frac{4}{\pi^2}Q(U), Q k ( V ♯ , k ( U )) ≤ s 2 18 π 2 ≤ π 2 4 Q ( U ) , Q k ( V p , k ( U ) ) = 24 ≥ 4 s π 2 Q ( U ) . Q_k(V^{p,k}(U))=24\geq\frac{4s}{\pi^2}Q(U). Q k ( V p , k ( U )) = 24 ≥ π 2 4 s Q ( U ) . Proof. Here ρ G = 1 \rho_G=1 ρ G = 1 S S S 3 s 3s 3 s Θ p = ± 2 T / s \Theta_p=\pm2T/s Θ p = ± 2 T / s 2 π / s \sqrt{2}\pi/s 2 π / s η 0 A = 6 π 2 / s ≤ 1 144 \eta_0A=6\pi^2/s\leq\frac{1}{144} η 0 A = 6 π 2 / s ≤ 144 1 s ≥ 864 π 2 s\geq864\pi^2 s ≥ 864 π 2 s ≥ 2 14 s\geq2^{14} s ≥ 2 14 A 2 = 18 π 2 A^2=18\pi^2 A 2 = 18 π 2 Q ( U ) = 24 s sin 2 ( π / 2 s ) Q(U)=24s\sin^2(\pi/2s) Q ( U ) = 24 s sin 2 ( π /2 s ) ( 6 π 2 / s ) ( 1 − π 2 / ( 12 s 2 ) ) (6\pi^2/s)(1-\pi^2/(12s^2)) ( 6 π 2 / s ) ( 1 − π 2 / ( 12 s 2 )) 6 π 2 / s 6\pi^2/s 6 π 2 / s
So on torus-valued fields of small curvature the cut-off average behaves like the linear averages: the curvature is averaged over the sixteen offsets at every step, the energy never increases, and a thin path is spread out and contracted instead of being carried to the coarse lattice (Figure 15 ).
Figure 15. (a) In the torus, the coarse plaquette of the cut-off average is the mean of the fluxes F 2 y + o F_{2y+o} F 2 y + o o o o k k k 24 24 24 18 π 2 / s 2 18\pi^2/s^2 18 π 2 / s 2 6 π 2 / s 6\pi^2/s 6 π 2 / s
For general fields the paths do not commute, and the logarithms carry commutators from the second order on. The next two lemmas bound these commutators by the square of the local curvature. The same composition then holds for every compact group, with a loss of at most 1 + O ( η j ) 1+O(\eta_j) 1 + O ( η j ) j j j
Lemma 104. Let Z 1 , … , Z n ∈ g Z_1,\ldots,Z_n\in\mathfrak{g} Z 1 , … , Z n ∈ g r = ∑ i ∥ Z i ∥ ≤ min { 1 4 , ρ G } r=\sum_i\lVert Z_i\rVert\leq\min\{\frac{1}{4},\rho_G\} r = ∑ i ∥ Z i ∥ ≤ min { 4 1 , ρ G } g = e Z 1 ⋯ e Z n g=e^{Z_1}\cdots e^{Z_n} g = e Z 1 ⋯ e Z n δ ( g , I ) ≤ r \delta(g,I)\leq r δ ( g , I ) ≤ r log g ∈ g \log g\in\mathfrak{g} log g ∈ g
∥ log g − ∑ i Z i ∥ ≤ 1.6 r 2 . \left\lVert\log g-\sum_i Z_i\right\rVert\leq1.6r^2. log g − i ∑ Z i ≤ 1.6 r 2 . Proof. The distance δ \delta δ δ ( g , I ) ≤ ∑ i δ ( e Z i , I ) ≤ r \delta(g,I)\leq\sum_i\delta(e^{Z_i},I)\leq r δ ( g , I ) ≤ ∑ i δ ( e Z i , I ) ≤ r log g ∈ g \log g\in\mathfrak{g} log g ∈ g ρ G \rho_G ρ G g − I − ∑ i Z i g-I-\sum_i Z_i g − I − ∑ i Z i e r − 1 − r ≤ 1 2 r 2 e r ≤ 0.643 r 2 e^r-1-r\leq\frac{1}{2}r^2e^r\leq0.643r^2 e r − 1 − r ≤ 2 1 r 2 e r ≤ 0.643 r 2 M = g − I M=g-I M = g − I ∥ M ∥ ≤ e r − 1 ≤ 1.137 r ≤ 0.285 \lVert M\rVert\leq e^r-1\leq1.137r\leq0.285 ∥ M ∥ ≤ e r − 1 ≤ 1.137 r ≤ 0.285 M M M 1 1 1 ∣ z − 1 ∣ < 1 \lvert z-1\rvert<1 ∣ z − 1 ∣ < 1 g g g log g = ∑ k ≥ 1 ( − 1 ) k + 1 M k / k \log g=\sum_{k\geq1}(-1)^{k+1}M^k/k log g = ∑ k ≥ 1 ( − 1 ) k + 1 M k / k
∥ log g − M ∥ ≤ ∥ M ∥ 2 2 ( 1 − ∥ M ∥ ) ≤ 0.905 r 2 , \lVert\log g-M\rVert\leq\frac{\lVert M\rVert^2}{2(1-\lVert M\rVert)}\leq0.905r^2, ∥ log g − M ∥ ≤ 2 ( 1 − ∥ M ∥) ∥ M ∥ 2 ≤ 0.905 r 2 , and the two bounds add up to less than 1.6 r 2 1.6r^2 1.6 r 2
Let U U U G G G δ ( U p , I ) ≤ ρ G \delta(U_p,I)\leq\rho_G δ ( U p , I ) ≤ ρ G Θ p = log U p \Theta_p=\log U_p Θ p = log U p Θ p \Theta_p Θ p P P P y y y { μ , ν } \{\mu,\nu\} { μ , ν } ℓ 1 = ( y , μ ) \ell_1=(y,\mu) ℓ 1 = ( y , μ ) ℓ 2 = ( y + e μ , ν ) \ell_2=(y+e_\mu,\nu) ℓ 2 = ( y + e μ , ν ) ℓ 3 = ( y + e ν , μ ) \ell_3=(y+e_\nu,\mu) ℓ 3 = ( y + e ν , μ ) ℓ 4 = ( y , ν ) \ell_4=(y,\nu) ℓ 4 = ( y , ν ) y i y_i y i o ∈ { 0 , 1 } 4 o\in\{0,1\}^4 o ∈ { 0 , 1 } 4 P i , o P_{i,o} P i , o ℓ i \ell_i ℓ i 2 y i + o 2y_i+o 2 y i + o
K P , o = P 1 , o P 2 , o P 3 , o − 1 P 4 , o − 1 , F ‾ P = 1 16 ∑ o log K P , o . K_{P,o}=P_{1,o}P_{2,o}P_{3,o}^{-1}P_{4,o}^{-1},\qquad\overline{F}_P=\frac{1}{16}\sum_o\log K_{P,o}. K P , o = P 1 , o P 2 , o P 3 , o − 1 P 4 , o − 1 , F P = 16 1 o ∑ log K P , o . The tree segments between consecutive sides cancel. Hence K P , o K_{P,o} K P , o □ 2 y + o \square_{2y+o} □ 2 y + o 2 y 2y 2 y log K P , o = F 2 y + o \log K_{P,o}=F_{2y+o} log K P , o = F 2 y + o F ‾ P \overline{F}_P F P σ P , o \sigma_{P,o} σ P , o ∥ Θ p ∥ \lVert\Theta_p\rVert ∥ Θ p ∥ □ 2 y + o \square_{2y+o} □ 2 y + o
Lemma 105. Let U U U G G G δ ( U p , I ) ≤ ρ G / 1000 \delta(U_p,I)\leq\rho_G/1000 δ ( U p , I ) ≤ ρ G /1000 ∥ Θ p ∥ ≤ ρ G / 1000 \lVert\Theta_p\rVert\leq\rho_G/1000 ∥ Θ p ∥ ≤ ρ G /1000 χ γ , μ ( U ) = 1 \chi_{\gamma,\mu}(U)=1 χ γ , μ ( U ) = 1 δ ( V ♯ ( U ) P , I ) ≤ 0.03 ρ G \delta(V^{\sharp}(U)_P,I)\leq0.03\rho_G δ ( V ♯ ( U ) P , I ) ≤ 0.03 ρ G
∥ log V ♯ ( U ) P − F ‾ P ∥ ≤ 3.2 r 2 , \lVert\log V^{\sharp}(U)_P-\overline{F}_P\rVert\leq3.2r^2, ∥ log V ♯ ( U ) P − F P ∥ ≤ 3.2 r 2 , ∥ F ‾ P ∥ ≤ 1 + 6.4 η 16 ∑ o σ P , o . (22) \lVert\overline{F}_P\rVert\leq\frac{1+6.4\eta}{16}\sum_o\sigma_{P,o}. \tag*{(22)} ∥ F P ∥ ≤ 16 1 + 6.4 η o ∑ σ P , o . (22) with r P ≤ 28.26 η r_P \le28.26\eta r P ≤ 28.26 η E E E A A A ∥ Θ p ∥ 2 \|\Theta_p\|^2 ∥ Θ p ∥ 2 ∥ Θ p ∥ \|\Theta_p\| ∥ Θ p ∥ E ′ E' E ′ A ′ A' A ′ η ′ \eta' η ′ ∥ log V P ♯ ( U ) ∥ \|\log V_P^\sharp(U)\| ∥ log V P ♯ ( U ) ∥
E ′ ≤ ( 1 + 6030 η ) E , A ′ ≤ ( 1 + 9147 η ) A 4 , \sqrt{E'} \le(1+6030\eta)\sqrt{E},\qquad A' \le(1+9147\eta)\frac{A}{4}, E ′ ≤ ( 1 + 6030 η ) E , A ′ ≤ ( 1 + 9147 η ) 4 A , η ′ ≤ 4 ( 1 + 647 η ) η . \eta' \le4(1+647\eta)\eta. η ′ ≤ 4 ( 1 + 647 η ) η . Proof. Paths. Fix a coarse link ( y , μ ) (y,\mu) ( y , μ ) x = 2 y + o x=2y+o x = 2 y + o P 2 y − 1 P x P_{2y}^{-1}P_x P 2 y − 1 P x h U p ± 1 h − 1 = exp ( ± h Θ p h − 1 ) hU_p^{\pm1}h^{-1}=\exp(\pm h\Theta_ph^{-1}) h U p ± 1 h − 1 = exp ( ± h Θ p h − 1 ) σ ( y , μ ) , o ≤ 6 η \sigma_{(y,\mu),o}\le6\eta σ ( y , μ ) , o ≤ 6 η ∥ Θ p ∥ \|\Theta_p\| ∥ Θ p ∥ δ ( P x , P 2 y ) ≤ 6 η ≤ ρ G / 2 \delta(P_x,P_{2y})\le6\eta\le\rho_G/2 δ ( P x , P 2 y ) ≤ 6 η ≤ ρ G /2 χ = 1 \chi=1 χ = 1 D ( y , μ ) , o = log ( P 2 y − 1 P x ) D_{(y,\mu),o}=\log(P_{2y}^{-1}P_x) D ( y , μ ) , o = log ( P 2 y − 1 P x ) g \mathfrak{g} g
∥ D ( y , μ ) , o ∥ ≤ ( 1 + 9.6 η ) σ ( y , μ ) , o . \|D_{(y,\mu),o}\|\le(1+9.6\eta)\sigma_{(y,\mu),o}. ∥ D ( y , μ ) , o ∥ ≤ ( 1 + 9.6 η ) σ ( y , μ ) , o . In the same way K P , o K_{P,o} K P , o ∥ log K P , o ∥ ≤ ( 1 + 6.4 η ) σ P , o \|\log K_{P,o}\|\le(1+6.4\eta)\sigma_{P,o} ∥ log K P , o ∥ ≤ ( 1 + 6.4 η ) σ P , o o o o (22) .
The plaquette. Write D i , o D_{i,o} D i , o D ℓ i , o D_{\ell_i,o} D ℓ i , o P i , o = P i , 0 e D i , o P_{i,o}=P_{i,0}e^{D_{i,o}} P i , o = P i , 0 e D i , o D i , 0 = 0 D_{i,0}=0 D i , 0 = 0 V P ♯ = P i , 0 e m i V_P^\sharp=P_{i,0}e^{m_i} V P ♯ = P i , 0 e m i m i = 1 16 ∑ o D i , o m_i=\frac{1}{16}\sum_oD_{i,o} m i = 16 1 ∑ o D i , o V ♯ V^\sharp V ♯ D i , o D_{i,o} D i , o m i m_i m i K P , o K_{P,o} K P , o V P ♯ V_P^\sharp V P ♯ P 1 , 0 = P 2 , 0 = P 3 , 0 = I P_{1,0}=P_{2,0}=P_{3,0}=I P 1 , 0 = P 2 , 0 = P 3 , 0 = I P 4 , 0 = K P , 0 − 1 P_{4,0}=K_{P,0}^{-1} P 4 , 0 = K P , 0 − 1 c = log K P , 0 c=\log K_{P,0} c = log K P , 0
V P ♯ = e m 1 e m 2 e − m 3 e − m 4 e c , V_P^\sharp=e^{m_1}e^{m_2}e^{-m_3}e^{-m_4}e^c, V P ♯ = e m 1 e m 2 e − m 3 e − m 4 e c , K P , o = e D 1 , o e D 2 , o e − D 3 , o e − D 4 , o e c . K_{P,o}=e^{D_{1,o}}e^{D_{2,o}}e^{-D_{3,o}}e^{-D_{4,o}}e^c. K P , o = e D 1 , o e D 2 , o e − D 3 , o e − D 4 , o e c . Put r P = ∑ i max o ∥ D i , o ∥ + ∥ c ∥ r_P=\sum_i\max_o\|D_{i,o}\|+\|c\| r P = ∑ i max o ∥ D i , o ∥ + ∥ c ∥ ∥ m i ∥ ≤ max o ∥ D i , o ∥ \|m_i\|\le\max_o\|D_{i,o}\| ∥ m i ∥ ≤ max o ∥ D i , o ∥ r P r_P r P
r P ≤ 24 ( 1 + 9.6 η ) η + 4 ( 1 + 6.4 η ) η ≤ 28.26 η . r_P\le24(1+9.6\eta)\eta+4(1+6.4\eta)\eta\le28.26\eta. r P ≤ 24 ( 1 + 9.6 η ) η + 4 ( 1 + 6.4 η ) η ≤ 28.26 η . Lemma 104 gives δ ( V P ♯ , I ) ≤ r P ≤ 0.03 ρ G \delta(V_P^\sharp,I)\le r_P\le0.03\rho_G δ ( V P ♯ , I ) ≤ r P ≤ 0.03 ρ G
log V P ♯ = m 1 + m 2 − m 3 − m 4 + c + R , \log V_P^\sharp=m_1+m_2-m_3-m_4+c+R, log V P ♯ = m 1 + m 2 − m 3 − m 4 + c + R , log K P , o = D 1 , o + D 2 , o − D 3 , o − D 4 , o + c + R o , \log K_{P,o}=D_{1,o}+D_{2,o}-D_{3,o}-D_{4,o}+c+R_o, log K P , o = D 1 , o + D 2 , o − D 3 , o − D 4 , o + c + R o , with ∥ R ∥ , ∥ R o ∥ ≤ 1.6 r P 2 \|R\|,\|R_o\|\le1.6r_P^2 ∥ R ∥ , ∥ R o ∥ ≤ 1.6 r P 2 o o o m i m_i m i D i , o D_{i,o} D i , o (22) .
Sums. Each fine plaquette lies in four squares □ x \square_x □ x ( P , o ) (P,o) ( P , o ) (22) and Cauchy–Schwarz therefore give ∑ P ∥ F ˉ P ∥ 2 ≤ ( 1 + 6.4 η ) 2 E \sum_P\|\bar F_P\|^2\le(1+6.4\eta)^2E ∑ P ∥ F ˉ P ∥ 2 ≤ ( 1 + 6.4 η ) 2 E ∑ P ∥ F ˉ P ∥ ≤ ( 1 + 6.4 η ) A / 4 \sum_P\|\bar F_P\|\le(1+6.4\eta)A/4 ∑ P ∥ F ˉ P ∥ ≤ ( 1 + 6.4 η ) A /4 U ℓ \mathcal U_\ell U ℓ ℓ = ( y , μ ) \ell=(y,\mu) ℓ = ( y , μ ) { μ , α } \{\mu,\alpha\} { μ , α } α ≠ μ \alpha\ne\mu α = μ μ \mu μ 2 y μ 2y_\mu 2 y μ 2 y μ + 1 2y_\mu+1 2 y μ + 1 2 y μ + 2 2y_\mu+2 2 y μ + 2 U ℓ \mathcal U_\ell U ℓ m ℓ ′ = ∑ p ∈ U ℓ ∥ Θ p ∥ m'_\ell=\sum_{p\in\mathcal U_\ell}\|\Theta_p\| m ℓ ′ = ∑ p ∈ U ℓ ∥ Θ p ∥
r P ≤ 1.0096 ( ∑ ℓ ⊂ P m ℓ ′ + σ P , 0 ) . r_P\le1.0096\left(\sum_{\ell\subset P}m'_\ell+\sigma_{P,0}\right). r P ≤ 1.0096 ( ℓ ⊂ P ∑ m ℓ ′ + σ P , 0 ) . A coarse link is a side of six coarse plaquettes, and a plaquette lies in □ 2 y \square_{2y} □ 2 y
∑ P r P ≤ 1.0096 ( 6 ⋅ 4 A + A ) ≤ 25.24 A , \sum_P r_P\le1.0096(6\cdot4A+A)\le25.24A, P ∑ r P ≤ 1.0096 ( 6 ⋅ 4 A + A ) ≤ 25.24 A , ∑ P r P 2 ≤ 1.0096 2 ⋅ 5 ( 6 ⋅ 36 ⋅ 4 E + 4 E ) ≤ 4424 E . \sum_P r_P^2\le1.0096^2\cdot5(6\cdot36\cdot4E+4E)\le4424E. P ∑ r P 2 ≤ 1.009 6 2 ⋅ 5 ( 6 ⋅ 36 ⋅ 4 E + 4 E ) ≤ 4424 E . Hence, with r P ≤ 28.26 η r_P\le28.26\eta r P ≤ 28.26 η
∑ P 3.2 r P 2 ≤ 3.2 ⋅ 28.26 ⋅ 25.24 η A ≤ 2283 η A , \sum_P3.2r_P^2\le3.2\cdot28.26\cdot25.24\eta A\le2283\eta A, P ∑ 3.2 r P 2 ≤ 3.2 ⋅ 28.26 ⋅ 25.24 η A ≤ 2283 η A , ( ∑ P ( 3.2 r P 2 ) 2 ) 1 / 2 ≤ 3.2 ⋅ 28.26 η ( 4424 E ) 1 / 2 ≤ 6015 η E . \left(\sum_P(3.2r_P^2)^2\right)^{1/2}\le3.2\cdot28.26\eta(4424E)^{1/2}\le6015\eta\sqrt{E}. ( P ∑ ( 3.2 r P 2 ) 2 ) 1/2 ≤ 3.2 ⋅ 28.26 η ( 4424 E ) 1/2 ≤ 6015 η E . The triangle inequality, for the Euclidean norm over P P P E ′ ≤ ( 1 + 6.4 η + 6015 η ) E \sqrt{E'}\le(1+6.4\eta+6015\eta)\sqrt{E} E ′ ≤ ( 1 + 6.4 η + 6015 η ) E A ′ ≤ ( 1 + 6.4 η ) A / 4 + 2283 η A A'\le(1+6.4\eta)A/4+2283\eta A A ′ ≤ ( 1 + 6.4 η ) A /4 + 2283 η A η ′ ≤ ( 1 + 6.4 η ) 4 η + 3.2 ( 28.26 η ) 2 ≤ 4 ( 1 + 647 η ) η \eta'\le(1+6.4\eta)4\eta+3.2(28.26\eta)^2\le4(1+647\eta)\eta η ′ ≤ ( 1 + 6.4 η ) 4 η + 3.2 ( 28.26 η ) 2 ≤ 4 ( 1 + 647 η ) η □ \square □
For a G G G k k k G G G V ♯ V^\sharp V ♯ ∥ Θ P ( j ) ∥ \|\Theta_P^{(j)}\| ∥ Θ P ( j ) ∥
Theorem 106. Let G ⊂ U ( r ) G\subset U(r) G ⊂ U ( r ) U U U G G G 2 k 2^k 2 k δ ( U P , I ) ≤ ρ G \delta(U_P,I)\le\rho_G δ ( U P , I ) ≤ ρ G η j \eta_j η j A j A_j A j E j E_j E j ∥ Θ P ( j ) ∥ \|\Theta_P^{(j)}\| ∥ Θ P ( j ) ∥ j j j
(i) If η j ≤ ρ G / 1000 \eta_j\le\rho_G/1000 η j ≤ ρ G /1000 0 ≤ j < k 0\le j<k 0 ≤ j < k σ = ∑ j < k η j \sigma=\sum_{j<k}\eta_j σ = ∑ j < k η j
E k ≤ e 12060 σ E 0 , A k ≤ e 9147 σ 4 − k A 0 , E_k\le e^{12060\sigma}E_0,\qquad A_k\le e^{9147\sigma}4^{-k}A_0, E k ≤ e 12060 σ E 0 , A k ≤ e 9147 σ 4 − k A 0 , η k ≤ e 647 σ 4 k η 0 . \eta_k\le e^{647\sigma}4^k\eta_0. η k ≤ e 647 σ 4 k η 0 . (ii) If η 0 A 0 ≤ 2 ⋅ 10 − 10 ρ G 2 \eta_0A_0\le2\cdot10^{-10}\rho_G^2 η 0 A 0 ≤ 2 ⋅ 1 0 − 10 ρ G 2 k k k σ ≤ 7.25 η 0 A 0 \sigma\le7.25\sqrt{\eta_0A_0} σ ≤ 7.25 η 0 A 0
Q k ( V ♯ , k ( U ) ) ≤ e 8.8 ⋅ 10 4 η 0 A 0 E 0 ≤ 1.0001 e 8.8 ⋅ 10 4 η 0 A 0 Q ( U ) , Q_k(V^{\sharp,k}(U))\le e^{8.8\cdot10^4\sqrt{\eta_0A_0}}E_0\le1.0001e^{8.8\cdot10^4\sqrt{\eta_0A_0}}Q(U), Q k ( V ♯ , k ( U )) ≤ e 8.8 ⋅ 1 0 4 η 0 A 0 E 0 ≤ 1.0001 e 8.8 ⋅ 1 0 4 η 0 A 0 Q ( U ) , Q k ( V ♯ , k ( U ) ) ≤ e 2 16 − k A 0 2 . Q_k(V^{\sharp,k}(U))\le e^2 16^{-k}A_0^2. Q k ( V ♯ , k ( U )) ≤ e 2 1 6 − k A 0 2 . Proof. (i) Apply Lemma 105 at each step to the field in the tree gauge; the gauge fixing does not change the norms. The lemma gives δ ( U P ( j + 1 ) , I ) ≤ ρ G \delta(U_P^{(j+1)},I)\le\rho_G δ ( U P ( j + 1 ) , I ) ≤ ρ G Θ ( j + 1 ) \Theta^{(j+1)} Θ ( j + 1 ) 1 + t ≤ e t 1+t\le e^t 1 + t ≤ e t
(ii) Suppose that for i < j i<j i < j η i ≤ ρ G / 1000 \eta_i\le\rho_G/1000 η i ≤ ρ G /1000 ∑ i ≤ j η i ≤ 7.25 η 0 A 0 ≤ 1 / 9747 \sum_{i\le j}\eta_i\le7.25\sqrt{\eta_0A_0}\le1/9747 ∑ i ≤ j η i ≤ 7.25 η 0 A 0 ≤ 1/9747 A j ≤ e 4 j − k A 0 A_j\le e^{4^{j-k}}A_0 A j ≤ e 4 j − k A 0 η j ≤ e 4 j η 0 \eta_j\le e^{4^j}\eta_0 η j ≤ e 4 j η 0 η j ≤ A j \eta_j\le A_j η j ≤ A j η j ≤ e min { 4 j η 0 , 4 j − k A 0 } ≤ e η 0 A 0 ≤ ρ G / 1000 \eta_j\le e^{\min\{4^j\eta_0,4^{j-k}A_0\}}\le e^{\sqrt{\eta_0A_0}}\le\rho_G/1000 η j ≤ e m i n { 4 j η 0 , 4 j − k A 0 } ≤ e η 0 A 0 ≤ ρ G /1000 i ∗ i_* i ∗ 4 i ∗ η 0 ≤ 4 − i ∗ A 0 4^{i_*}\eta_0\le4^{-i_*}A_0 4 i ∗ η 0 ≤ 4 − i ∗ A 0 min { 4 j η 0 , 4 j − k A 0 } \min\{4^j\eta_0,4^{j-k}A_0\} min { 4 j η 0 , 4 j − k A 0 } i ∗ i_* i ∗ i ∗ i_* i ∗ i ∗ + 1 i_*+1 i ∗ + 1 η 0 A 0 \sqrt{\eta_0A_0} η 0 A 0 i i i 8 3 η 0 A 0 \frac{8}{3}\sqrt{\eta_0A_0} 3 8 η 0 A 0 ∑ i ≤ j η i ≤ 8 e 3 η 0 A 0 ≤ 7.25 η 0 A 0 \sum_{i\le j}\eta_i\le\frac{8e}{3}\sqrt{\eta_0A_0}\le7.25\sqrt{\eta_0A_0} ∑ i ≤ j η i ≤ 3 8 e η 0 A 0 ≤ 7.25 η 0 A 0 E k ≤ e 12060 ⋅ 7.25 η 0 A 0 E 0 E_k \le e^{12060\cdot7.25\sqrt{\eta_0 A_0}E_0} E k ≤ e 12060 ⋅ 7.25 η 0 A 0 E 0 Q k ( V ♯ , k ( U ) ) ≤ E k Q_k(V^{\sharp,k}(U)) \le E_k Q k ( V ♯ , k ( U )) ≤ E k E k ≤ η k A k ≤ A 2 2 16 − k A 0 2 E_k \le\eta_k A_k \le A_2^2 16^{-k}A_0^2 E k ≤ η k A k ≤ A 2 2 1 6 − k A 0 2 i Θ p i\Theta_p i Θ p Θ p \Theta_p Θ p ∣ θ ∣ ≤ η 0 ≤ η 0 A 0 < 10 − 4 |\theta| \le\eta_0 \le\sqrt{\eta_0 A_0}<10^{-4} ∣ θ ∣ ≤ η 0 ≤ η 0 A 0 < 1 0 − 4 θ 2 ≤ 1.0001 ⋅ 4 sin 2 ( θ / 2 ) \theta^2 \le1.0001\cdot4\sin^2(\theta/2) θ 2 ≤ 1.0001 ⋅ 4 sin 2 ( θ /2 )
For a torus Theorem 102 is sharper, and its hypothesis is weaker; the content of Theorem 106 is the non-abelian case. Neither theorem gives the supremum over boundary conditions in (D). Theorem 100 has the constants of one step. Theorems 102 and 106 need the curvature to be small at every scale: in Theorem 106 the losses e 6030 η j e^{6030\eta_j} e 6030 η j ℓ \ell ℓ ℓ \ell ℓ
Large fields under the Wilson measure
Theorem 106 needs every plaquette to be small. This subsection shows that, under the SU(2) Wilson measure, the plaquettes that are not small form clusters of microscopic size, and that the cut-off average departs from the average only next to them. It also shows that such plaquettes are present along the one-loop tuning, so the composition has to pass through them.
Let Λ L = ( Z / L Z ) 4 \Lambda_L=(\mathbb{Z}/L\mathbb{Z})^4 Λ L = ( Z / L Z ) 4 L = 2 m ≥ 4 L=2^m\ge4 L = 2 m ≥ 4 μ L \mu_L μ L Λ L \Lambda_L Λ L e − β W S ( U ) e^{-\beta_W S(U)} e − β W S ( U )
S ( U ) = ∑ p s p , s p = 1 − 1 2 Tr U p = 1 4 δ ( U p , I ) 2 . S(U)=\sum_p s_p,\qquad s_p=1-\frac{1}{2}\operatorname{Tr}U_p=\frac{1}{4}\delta(U_p,I)^2. S ( U ) = p ∑ s p , s p = 1 − 2 1 Tr U p = 4 1 δ ( U p , I ) 2 . This is the Wilson measure of § 2 on the torus, without gauge fixing; it has 4 L 4 4L^4 4 L 4 6 L 4 6L^4 6 L 4 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 bad if δ ( U p , I ) > ρ \delta(U_p,I)>\rho δ ( U p , I ) > ρ C x = x + [ 0 , 1 ] 4 C_x=x+[0,1]^4 C x = x + [ 0 , 1 ] 4 B x B_x B x
Lemma 107. Let Z L Z_L Z L μ L \mu_L μ L C 0 = 1 2 + 2 3 log ( 24 π 3 ) C_0=\frac{1}{2}+\frac{2}{3}\log(24\pi^3) C 0 = 2 1 + 3 2 log ( 24 π 3 ) β W ≥ 1 \beta_W\ge1 β W ≥ 1
− log Z L ≤ 6 L 4 ( log β W + C 0 ) , E μ L S ≤ 6 L 4 log β W + C 0 β W . -\log Z_L\le6L^4(\log\beta_W+C_0),\qquad\mathbb{E}_{\mu_L}S\le6L^4\frac{\log\beta_W+C_0}{\beta_W}. − log Z L ≤ 6 L 4 ( log β W + C 0 ) , E μ L S ≤ 6 L 4 β W log β W + C 0 . Proof. Write U = cos θ I + sin θ J U=\cos\theta I+\sin\theta J U = cos θ I + sin θ J 0 ≤ θ ≤ π 0\le\theta\le\pi 0 ≤ θ ≤ π J 2 = − I J^2=-I J 2 = − I θ \theta θ g − 1 h g^{-1}h g − 1 h θ \theta θ 2 π sin 2 θ \frac{2}{\pi}\sin^2\theta π 2 sin 2 θ r = ( 4 β W ) − 1 r=(4\sqrt{\beta_W})^{-1} r = ( 4 β W ) − 1 r r r 4 r ≤ 1 4r\le1 4 r ≤ 1 s p ≤ ( 4 r ) 2 / 2 = 1 / ( 2 β W ) s_p\le(4r)^2/2=1/(2\beta_W) s p ≤ ( 4 r ) 2 /2 = 1/ ( 2 β W ) S ≤ 3 L 4 S\le3L^4 S ≤ 3 L 4 r r r
2 π ∫ 0 r sin 2 t d t ≥ 8 r 3 3 π 3 = 1 24 π 3 β W 3 / 2 , \frac{2}{\pi}\int_0^r\sin^2t\,dt\ge\frac{8r^3}{3\pi^3}=\frac{1}{24\pi^3\beta_W^{3/2}}, π 2 ∫ 0 r sin 2 t d t ≥ 3 π 3 8 r 3 = 24 π 3 β W 3/2 1 , by sin t ≥ 2 t / π \sin t\ge2t/\pi sin t ≥ 2 t / π Z L ≥ e − 3 L 4 ( 24 π 3 β W 3 / 2 ) − 4 L 4 Z_L\ge e^{-3L^4}(24\pi^3\beta_W^{3/2})^{-4L^4} Z L ≥ e − 3 L 4 ( 24 π 3 β W 3/2 ) − 4 L 4 F ( b ) = − log Z L F(b)=-\log Z_L F ( b ) = − log Z L b b b F ( 0 ) = 0 F(0)=0 F ( 0 ) = 0 F ′ ( b ) = E S F'(b)=\mathbb{E}S F ′ ( b ) = E S b F ′ ( b ) ≤ F ( b ) bF'(b)\le F(b) b F ′ ( b ) ≤ F ( b )
Theorem 108. For β W ≥ 1 \beta_W\ge1 β W ≥ 1 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 A A A
μ L ( ⋂ x ∈ A B x ) ≤ q ∣ A ∣ , \mu_L\left(\bigcap_{x\in A}B_x\right)\le q^{|A|}, μ L ( x ∈ A ⋂ B x ) ≤ q ∣ A ∣ , where
q = q ρ ( β W ) = min { 1 , exp ( 6 ( log β W + C 0 ) − β W ρ 2 / 16 ) } . q=q_\rho(\beta_W)=\min\left\{1,\exp\left(6(\log\beta_W+C_0)-\beta_W\rho^2/16\right)\right\}. q = q ρ ( β W ) = min { 1 , exp ( 6 ( log β W + C 0 ) − β W ρ 2 /16 ) } . Proof. The measure μ L \mu_L μ L [64 ] : the plaquettes are either in the hyperplane or on one side of it, so conditionally on the links in the hyperplane the weight factors into two mirror images, and reflection preserves Haar measure and Tr U p \operatorname{Tr}U_p Tr U p 2 m 2^m 2 m [35 ] , [20 Section 5.3] :
μ L ( ⋂ x ∈ A B x ) ≤ μ L ( ⋂ x ∈ Λ L B x ) ∣ A ∣ / L 4 , \mu_L\left(\bigcap_{x\in A}B_x\right)\le\mu_L\left(\bigcap_{x\in\Lambda_L}B_x\right)^{|A|/L^4}, μ L ( x ∈ A ⋂ B x ) ≤ μ L ( x ∈ Λ L ⋂ B x ) ∣ A ∣/ L 4 , since the reflection of B x B_x B x B y B_y B y L 4 / 4 L^4/4 L 4 /4 S ≥ ρ 2 L 4 / 16 S\ge\rho^2L^4/16 S ≥ ρ 2 L 4 /16 e − β W ρ 2 L 4 / 16 e^{-\beta_W\rho^2L^4/16} e − β W ρ 2 L 4 /16 Z L Z_L Z L
Corollary 109. Join two unit cubes when their corners are at ℓ ∞ \ell^\infty ℓ ∞ r ≥ 1 r\ge1 r ≥ 1
μ L { some connected set of r bad cubes } ≤ L 4 ( 6400 q ) r . \mu_L\{\text{some connected set of }r\text{ bad cubes}\}\le L^4(6400q)^r. μ L { some connected set of r bad cubes } ≤ L 4 ( 6400 q ) r . Hence, on a torus of fixed physical side L a La L a a → 0 a\to0 a → 0 L = 2 m L=2^m L = 2 m β W ( a ) → ∞ \beta_W(a)\to\infty β W ( a ) → ∞ ρ \rho ρ
Proof. A connected set of r r r 2 ( r − 1 ) 2(r-1) 2 ( r − 1 ) 80 2 ( r − 1 ) ≤ 6400 r 80^{2(r-1)}\le6400^r 8 0 2 ( r − 1 ) ≤ 640 0 r Theorem 108 and sum over the L 4 L^4 L 4 R R R ⌈ R / a ⌉ \lceil R/a\rceil ⌈ R / a ⌉ 6400 q ≤ 1 2 6400q\le\frac12 6400 q ≤ 2 1 ( L a / a ) 4 − R / a → 0 (La/a)^{4- R/a}\to0 ( L a / a ) 4 − R / a → 0
The bad cubes control where the cut-off average differs from the average. For a coarse link ( y , μ ) (y,\mu) ( y , μ ) X y , μ \mathcal{X}_{y,\mu} X y , μ 2 y + t e μ + w 2y+t e_\mu+w 2 y + t e μ + w t ∈ { 0 , 1 , 2 } t\in\{0,1,2\} t ∈ { 0 , 1 , 2 } w ∈ { 0 , 1 } 4 w\in\{0,1\}^4 w ∈ { 0 , 1 } 4 w μ = 0 w_\mu=0 w μ = 0 Figure 16 ).
Figure 16. Large fields under the SU(2) Wilson measure. (a) A plane section of the lattice: bad unit cubes (red) form clusters, and the cut-off average can differ from the path average only on coarse links (orange) whose 24 cubes Z y , μ \mathcal{Z}_{y,\mu} Z y , μ e e e p 0 p_0 p 0 I I I − I -I − I U e U_e U e − I -I − I p 0 p_0 p 0 q ρ ( β W ) q_\rho(\beta_W) q ρ ( β W ) Theorem 108 for three values of ρ \rho ρ
Corollary 110. Let ρ = 1 12 \rho=\frac{1}{12} ρ = 12 1 X y , μ \mathcal{X}_{y,\mu} X y , μ χ y , μ ( U ) = 1 \chi_{y,\mu}(U)=1 χ y , μ ( U ) = 1 V y , μ ♯ V^\sharp_{y,\mu} V y , μ ♯ P 2 y P_{2y} P 2 y X y , μ \mathcal{X}_{y,\mu} X y , μ χ < 1 \chi<1 χ < 1 4.5 ⋅ 10 5 ( log β W + C 0 ) / β W 4.5\cdot10^5(\log\beta_W+C_0)/\beta_W 4.5 ⋅ 1 0 5 ( log β W + C 0 ) / β W 2 a 2a 2 a R > 0 R>0 R > 0
Proof. As in the proof of Lemma 105, P 2 y − 1 P x P_{2y}^{-1}P_x P 2 y − 1 P x { μ , α } \{\mu,\alpha\} { μ , α } 2 y + t e μ + w 2y+t e_\mu+w 2 y + t e μ + w δ ( P x , P 2 y ) ≤ 6 ρ = 1 2 \delta(P_x,P_{2y})\le6\rho=\frac{1}{2} δ ( P x , P 2 y ) ≤ 6 ρ = 2 1 χ = 1 \chi=1 χ = 1 z z z Z y , μ \mathcal{Z}_{y,\mu} Z y , μ z μ − 2 y μ ∈ { 0 , 1 , 2 } z_\mu-2y_\mu\in\{0,1,2\} z μ − 2 y μ ∈ { 0 , 1 , 2 } z α − 2 y α ∈ { 0 , 1 } z_\alpha-2y_\alpha\in\{0,1\} z α − 2 y α ∈ { 0 , 1 } α ≠ μ \alpha\ne\mu α = μ L 4 / 4 L^4/4 L 4 /4 s p > ρ 2 / 4 s_p>\rho^2/4 s p > ρ 2 /4 4 ⋅ 8 ⋅ 4 ⋅ 24 ⋅ 144 ( log β W + C 0 ) / β W 4\cdot8\cdot4\cdot24\cdot144(\log\beta_W+C_0)/\beta_W 4 ⋅ 8 ⋅ 4 ⋅ 24 ⋅ 144 ( log β W + C 0 ) / β W □ \square □
Small plaquettes everywhere, which Theorem 106 assumes, are nevertheless improbable along the one-loop tuning. For SU(2), β W = 4 g − 2 \beta_W=4g^{-2} β W = 4 g − 2 a = 1 / L a=1/L a = 1/ L β W = 8 b 0 log L + O ( 1 ) = 11 3 π 2 log L + O ( 1 ) \beta_W=8b_0\log L+O(1)=\frac{11}{3\pi^2}\log L+O(1) β W = 8 b 0 log L + O ( 1 ) = 3 π 2 11 log L + O ( 1 )
Proposition 111. Let 0 < ρ ≤ 1 0<\rho\le1 0 < ρ ≤ 1 v ∗ = 2 π ∫ arccos ( 3 / 4 ) π / 3 sin 2 t d t v_*=\frac{2}{\pi}\int_{\arccos(3/4)}^{\pi/3}\sin^2t\,dt v ∗ = π 2 ∫ a r c c o s ( 3/4 ) π /3 sin 2 t d t p = v ∗ e − 3 β W p=v_*e^{-3\beta_W} p = v ∗ e − 3 β W e e e p p p
μ L { no plaquette is bad } ≤ e − ⌊ L / 3 ⌋ 4 p , \mu_L\{ \text{no plaquette is bad} \}\le e^{-\lfloor L/3\rfloor^4p}, μ L { no plaquette is bad } ≤ e − ⌊ L /3 ⌋ 4 p , which tends to 0 0 0 β W ≤ B log L + O ( 1 ) \beta_W\le B\log L+O(1) β W ≤ B log L + O ( 1 ) B < 4 / 3 B<4/3 B < 4/3
Proof. Identify SU(2) with the unit quaternions. Conditionally on the other links, U e U_e U e e κ h ⋅ u e^{\kappa h\cdot u} e κh ⋅ u 0 ≤ κ ≤ 6 β W 0\le\kappa\le6\beta_W 0 ≤ κ ≤ 6 β W ∣ h ∣ = 1 |h|=1 ∣ h ∣ = 1 β W \beta_W β W U e U_e U e h h h I I I h = I h=I h = I h = I h=I h = I κ \kappa κ Tr U \operatorname{Tr}U Tr U 1 2 Tr U \frac{1}{2}\operatorname{Tr}U 2 1 Tr U arccos ( 1 − ρ 2 / 4 ) ≤ arccos ( 3 / 4 ) \arccos(1-\rho^2/4)\le\arccos(3/4) arccos ( 1 − ρ 2 /4 ) ≤ arccos ( 3/4 ) e 6 β W cos t sin 2 t e^{6\beta_W\cos t}\sin^2t e 6 β W c o s t sin 2 t p p p [ arccos ( 3 / 4 ) , π / 3 ] [\arccos(3/4),\pi/3] [ arccos ( 3/4 ) , π /3 ] cos t ≥ 1 2 \cos t\ge\frac{1}{2} cos t ≥ 2 1 e 6 β W π / 2 e^{6\beta_W}\pi/2 e 6 β W π /2
Let M = ⌊ L / 3 ⌋ M=\lfloor L/3\rfloor M = ⌊ L /3 ⌋ 3 z 3z 3 z z ∈ { 0 , … , M − 1 } 4 z\in\{0,\ldots,M-1\}^4 z ∈ { 0 , … , M − 1 } 4 { 0 , 1 } \{0,1\} { 0 , 1 } ( 1 − p ) M 4 (1-p)^{M^4} ( 1 − p ) M 4 β W = B log L + O ( 1 ) \beta_W=B\log L+O(1) β W = B log L + O ( 1 ) L 4 − 3 B L^{4-3B} L 4 − 3 B □ \square □
The joint bound of Theorem 108 is not a conditional one, and (D) needs a supremum over boundary conditions. The two differ genuinely. Fix a link e e e p 0 p_0 p 0 e e e I I I p 0 p_0 p 0 − I -I − I U e U_e U e e − 2 β W Tr U e e^{-2\beta_W\operatorname{Tr}U_e} e − 2 β W Tr U e p 0 p_0 p 0 U e ± 1 U_e^{\pm1} U e ± 1 β W → ∞ \beta_W\to\infty β W → ∞ ρ < 2 2 \rho<2\sqrt{2} ρ < 2 2
Acknowledgements
This paper rests on the work of many earlier scientists. Lattice gauge theory is due to Wilson [77 ] and its Hamiltonian form to Kogut and Susskind [51 ] ; reflection positivity and the reconstruction of the transfer matrix to Osterwalder and Schrader [62 , 63 ] , and their lattice form, with the first proof of a gap at strong coupling, to Osterwalder and Seiler [64 , 68 ] . The uniqueness method is Dobrushin’s [29 , 30 , 31 ] , the curvature method is that of Bakry and Émery [3 , 4 ] in the form of Shen, Zhu and Zhu [69 ] , the gauge fixing on a forest is Creutz’s [26 ] , and the curvature of Lie groups is Milnor’s [60 ] ; the localisation estimate of Sec. XII A is that of Combes and Thomas [27 ] . The analysis of the single-link law uses results of Barta [9 ] , Biernacki and Krzyż [19 ] , McShane [59 ] , Stein and Weiss [72 ] and Watson [76 ] , and the continuum problem is the one formulated by Jaffe and Witten [47 ] and studied by Bałaban [5 , 6 , 7 ] . The Yang–Mills manuscripts of Sussman and collaborators [73 , 74 ] were read in the preparation of § 2.6 .
Data and code availability
The proofs are analytic except for the evaluations in Propositions 34 and 35 and Lemma 38 and the numerical bounds of Lemma 16, which are computations in interval arithmetic with rigorous error bounds. The programs of these steps, in Python with python-flint (FLINT and Arb) and a C kernel for products of Taylor polynomials, their outputs and the list of certified boxes are in the repository https://github.com/creelie/4Y-M , release v4.8.0, together with a Lean 4 certificate of the line field and independent exact or numerical checks of the other finite statements.
The computations, the checks and parts of the drafting were done with AI assistance (Claude Opus 5.5, Claude Fable 5.1, GPT-6.1 Sol Ultra and GPT-6 Astra Ultra) under the direction of Deep Bhattacharjee. The combination of the forest gauge with the sharp quarter behind Theorem 1 was first assembled by Anaxia, the reasoning engine of the BLLM system [12 , 13 ] ; the line field, the free-boundary limit, the ceiling, the curvature bound, the SU(3) part, the plaquette blocks and the certified constants are the authors’ own. All three authors approved the final text.
Declarations
Funding. None. Competing interests. None. Ethics approval. Not applicable: the work involves no human participants, animals or personal data.
A The line field
Table III lists, for each residue class of x x x σ ( x ) \sigma(x) σ ( x ) π x ( μ ) = σ ( x + e μ ) \pi_x(\mu)=\sigma(x+e_\mu) π x ( μ ) = σ ( x + e μ ) π x \pi_x π x σ ( x ) \sigma(x) σ ( x )
B The comparison lemma
Let ( G , δ ) (G,\delta) ( G , δ ) D D D Ω = G A \Omega=G^A Ω = G A C e e ′ C_{ee'} C e e ′ § 2 , with row and column sums at most α < 1 \alpha<1 α < 1 C e e ′ = 0 C_{ee'}=0 C e e ′ = 0 e , e ′ e,e' e , e ′ ∣ e − e ′ ∣ ≤ 1 \lvert e-e'\rvert\leq1 ∣ e − e ′ ∣ ≤ 1 λ > 0 \lambda>0 λ > 0 θ = α e λ < 1 \theta=\alpha e^\lambda<1 θ = α e λ < 1
Lemma 112. Let S ⊂ A S\subset A S ⊂ A μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 Ω \Omega Ω γ e F \gamma_e^F γ e F e ∉ S e\notin S e ∈ / S T ⊂ A ∖ S T\subset A\setminus S T ⊂ A ∖ S w b = e − λ dist ( b , T ) w_b=e^{-\lambda\operatorname{dist}(b,T)} w b = e − λ dist ( b , T ) ∑ z ∈ S w z < ∞ \sum_{z\in S}w_z<\infty ∑ z ∈ S w z < ∞ μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 ρ ( b ) = E ( δ ( σ b , σ b ′ ) ) \rho(b)=\mathbb{E}(\delta(\sigma_b,\sigma'_b)) ρ ( b ) = E ( δ ( σ b , σ b ′ ))
∑ b ∉ S w b ρ ( b ) ≤ θ D 1 − θ ∑ z ∈ S w z . \sum_{b\notin S}w_b\rho(b)\leq\frac{\theta D}{1-\theta}\sum_{z\in S}w_z. b ∈ / S ∑ w b ρ ( b ) ≤ 1 − θ θ D z ∈ S ∑ w z . If S = ∅ S=\varnothing S = ∅ μ 1 = μ 2 \mu_1=\mu_2 μ 1 = μ 2
Proof. The couplings form a weakly compact set and Ψ ( π ) = ∑ b w b E π δ ( σ b , σ b ′ ) \Psi(\pi)=\sum_b w_b\mathbb{E}_\pi\delta(\sigma_b,\sigma'_b) Ψ ( π ) = ∑ b w b E π δ ( σ b , σ b ′ ) ∑ b w b < ∞ \sum_b w_b<\infty ∑ b w b < ∞ π ∗ \pi^* π ∗ e ∉ S e\notin S e ∈ / S ( σ e , σ e ′ ) (\sigma_e,\sigma'_e) ( σ e , σ e ′ ) γ e F ( ⋅ ∣ σ ) \gamma_e^F(\cdot\mid\sigma) γ e F ( ⋅ ∣ σ ) γ e F ( ⋅ ∣ σ ′ ) \gamma_e^F(\cdot\mid\sigma') γ e F ( ⋅ ∣ σ ′ ) [75 Cor. 5.22] . Both μ i \mu_i μ i γ e F \gamma_e^F γ e F W 1 ≤ ∑ e ′ C e e ′ δ ( σ e ′ , σ e ′ ′ ) W_1\leq\sum_{e'}C_{ee'}\delta(\sigma_{e'},\sigma'_{e'}) W 1 ≤ ∑ e ′ C e e ′ δ ( σ e ′ , σ e ′ ′ ) ρ ( e ) ≤ ∑ e ′ C e e ′ ρ ( e ′ ) \rho(e)\leq\sum_{e'}C_{ee'}\rho(e') ρ ( e ) ≤ ∑ e ′ C e e ′ ρ ( e ′ ) w e w_e w e e ∉ S e\notin S e ∈ / S C e e ′ ≠ 0 C_{ee'}\ne0 C e e ′ = 0 w e ≤ e λ w e ′ w_e\leq e^\lambda w_{e'} w e ≤ e λ w e ′
∑ e ∉ S w e ρ ( e ) ≤ θ [ ∑ e ′ ∉ S w e ′ ρ ( e ′ ) + ∑ z ∈ S w z ρ ( z ) ] , \sum_{e\notin S}w_e\rho(e)\leq\theta\left[\sum_{e'\notin S}w_{e'}\rho(e')+\sum_{z\in S}w_z\rho(z)\right], e ∈ / S ∑ w e ρ ( e ) ≤ θ e ′ ∈ / S ∑ w e ′ ρ ( e ′ ) + z ∈ S ∑ w z ρ ( z ) , all sums are finite, and ρ ≤ D \rho\leq D ρ ≤ D S S S S = ∅ S=\varnothing S = ∅ ρ = 0 \rho=0 ρ = 0
Corollary 113. Let μ \mu μ γ F \gamma^F γ F F F F S F S_F S F L F L_F L F H H H S H S_H S H S F S_F S F
∣ Cov μ ( F , H ) ∣ ≤ ∥ H ∥ ∞ L F θ D 1 − θ ∣ S H ∣ e − λ dist ( S F , S H ) . \left|\operatorname{Cov}_\mu(F,H)\right|\leq\lVert H\rVert_\infty L_F\frac{\theta D}{1-\theta}|S_H|e^{-\lambda\operatorname{dist}(S_F,S_H)}. ∣ Cov μ ( F , H ) ∣ ≤ ∥ H ∥ ∞ L F 1 − θ θ D ∣ S H ∣ e − λ dist ( S F , S H ) . Proof. Condition on the links of S H S_H S H [50 Ch. 3] ; the conditional measures μ σ = μ ( ⋅ ∣ σ S H ) \mu^\sigma=\mu(\cdot\mid\sigma_{S_H}) μ σ = μ ( ⋅ ∣ σ S H ) γ e F \gamma_e^F γ e F e ∉ S H e\notin S_H e ∈ / S H S = S H S=S_H S = S H T = S F T=S_F T = S F w b = 1 w_b=1 w b = 1 S F S_F S F w z ≤ e − λ dist ( S F , S H ) w_z\le e^{-\lambda\operatorname{dist}(S_F,S_H)} w z ≤ e − λ dist ( S F , S H ) S H S_H S H ∣ μ σ ( F ) − μ σ ′ ( F ) ∣ ≤ L F ∑ b ∈ S F ρ ( b ) ≤ L F θ D 1 − θ ∣ S H ∣ e − λ dist ( S F , S H ) |\mu^\sigma(F)-\mu^{\sigma'}(F)|\le L_F\sum_{b\in S_F}\rho(b)\le L_F\frac{\theta D}{1-\theta}|S_H|e^{-\lambda\operatorname{dist}(S_F,S_H)} ∣ μ σ ( F ) − μ σ ′ ( F ) ∣ ≤ L F ∑ b ∈ S F ρ ( b ) ≤ L F 1 − θ θ D ∣ S H ∣ e − λ dist ( S F , S H ) H H H σ S H \sigma_{S_H} σ S H Cov μ ( F , H ) = ∬ H ( σ ) ( μ σ ( F ) − μ σ ′ ( F ) ) d μ ( σ ) d μ ( σ ′ ) \operatorname{Cov}_\mu(F,H)=\iint H(\sigma)(\mu^\sigma(F)-\mu^{\sigma'}(F))\,d\mu(\sigma)d\mu(\sigma') Cov μ ( F , H ) = ∬ H ( σ ) ( μ σ ( F ) − μ σ ′ ( F )) d μ ( σ ) d μ ( σ ′ )
C The plaquette blocks
Lemma 114. Let the blocks partition A A A C B j C_{Bj} C B j ∑ B C B j ≤ α < 1 \sum_B C_{Bj}\le\alpha<1 ∑ B C B j ≤ α < 1 j j j C B j = 0 C_{Bj}=0 C B j = 0 B B B j j j λ > 0 \lambda>0 λ > 0 θ = α e 2 λ < 1 \theta=\alpha e^{2\lambda}<1 θ = α e 2 λ < 1 S ⊂ A S\subset A S ⊂ A μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 γ B F \gamma_B^F γ B F B ⊄ S B\not\subset S B ⊂ S T T T w b w_b w b ∑ z ∈ S w z < ∞ \sum_{z\in S}w_z<\infty ∑ z ∈ S w z < ∞ ∑ b ∉ S w b ρ ( b ) ≤ θ D 1 − θ ∑ z ∈ S w z \sum_{b\notin S}w_b\rho(b)\le\frac{\theta D}{1-\theta}\sum_{z\in S}w_z ∑ b ∈ / S w b ρ ( b ) ≤ 1 − θ θ D ∑ z ∈ S w z μ 1 = μ 2 \mu_1=\mu_2 μ 1 = μ 2 S = ∅ S=\varnothing S = ∅
Proof. Take a minimiser π ∗ \pi^* π ∗ Ψ \Psi Ψ B ⊄ S B\not\subset S B ⊂ S ( σ B , σ B ′ ) (\sigma_B,\sigma'_B) ( σ B , σ B ′ ) d d d d d d ∑ j C B j ρ ( j ) \sum_j C_{Bj}\rho(j) ∑ j C B j ρ ( j ) w ‾ B = max i ∈ B w i \overline w_B=\max_{i\in B}w_i w B = max i ∈ B w i
∑ i ∈ B w i ρ ( i ) ≤ w ‾ B ∑ j C B j ρ ( j ) . \sum_{i\in B}w_i\rho(i)\le\overline w_B\sum_j C_{Bj}\rho(j). i ∈ B ∑ w i ρ ( i ) ≤ w B j ∑ C B j ρ ( j ) . and w ‾ B ≤ e 2 λ w j \overline w_B\le e^{2\lambda}w_j w B ≤ e 2 λ w j C B j ≠ 0 C_{Bj}\ne0 C B j = 0 S S S
∑ b ∉ S w b ρ ( b ) ≤ θ [ ∑ j ∉ S w j ρ ( j ) + ∑ z ∈ S w z ρ ( z ) ] , \sum_{b\notin S}w_b\rho(b)\le\theta\left[\sum_{j\notin S}w_j\rho(j)+\sum_{z\in S}w_z\rho(z)\right], b ∈ / S ∑ w b ρ ( b ) ≤ θ j ∈ / S ∑ w j ρ ( j ) + z ∈ S ∑ w z ρ ( z ) , which gives the claim as in Lemma 112.
Lemma 115. Let n ≥ 2 n\ge2 n ≥ 2 V x V_x V x x ∈ 2 Z 4 x\in2\mathbb{Z}^4 x ∈ 2 Z 4 D ( n , α ) D(n,\alpha) D ( n , α )
(a) If a probability measure μ \mu μ G A G^A G A γ e F \gamma_e^F γ e F e e e V V V γ V \gamma_V γ V
(b) Fix λ > 0 \lambda>0 λ > 0 θ = α e λ ρ n < 1 \theta=\alpha e^{\lambda\rho_n}<1 θ = α e λ ρ n < 1 S ⊂ A S\subset A S ⊂ A μ 1 , μ 2 \mu_1,\mu_2 μ 1 , μ 2 γ V x \gamma_{V_x} γ V x x x x V x ∩ S = ∅ V_x\cap S=\varnothing V x ∩ S = ∅ T T T w b w_b w b ∑ z ∈ S w z < ∞ \sum_{z\in S}w_z<\infty ∑ z ∈ S w z < ∞
∑ b ∉ S w b ρ ( b ) ≤ D ( 1 + c n e λ ρ n ) 1 − θ ∑ z ∈ S w z , \sum_{b\notin S}w_b\rho(b)\le\frac{D(1+c_ne^{\lambda\rho_n})}{1-\theta}\sum_{z\in S}w_z, b ∈ / S ∑ w b ρ ( b ) ≤ 1 − θ D ( 1 + c n e λ ρ n ) z ∈ S ∑ w z , and μ 1 = μ 2 \mu_1=\mu_2 μ 1 = μ 2 S = ∅ S=\varnothing S = ∅
Proof. (a) Let ν η \nu_\eta ν η Q V Q_V Q V η \eta η V V V f f f G V G^V G V g g g η \eta η γ e F ( f g ) = g γ e F f \gamma_e^F(fg)=g\gamma_e^Ff γ e F ( f g ) = g γ e F f e ∈ V e\in V e ∈ V ∫ g ( η ) ( ν η ( γ e F f ) − ν η ( f ) ) d μ = 0 \int g(\eta)(\nu_\eta(\gamma_e^Ff)-\nu_\eta(f))\,d\mu=0 ∫ g ( η ) ( ν η ( γ e F f ) − ν η ( f )) d μ = 0 g g g f f f ν η \nu_\eta ν η γ e F ( ⋅ ∣ ⋅ , η ) \gamma_e^F(\cdot\mid\cdot,\eta) γ e F ( ⋅ ∣ ⋅ , η ) e ∈ V e\in V e ∈ V μ \mu μ η \eta η γ V ( ⋅ ∣ η ) \gamma_V(\cdot\mid\eta) γ V ( ⋅ ∣ η ) γ e F \gamma_e^F γ e F P η P_\eta P η γ e F \gamma_e^F γ e F e ∈ V e\in V e ∈ V G V G^V G V e − 12 N 2 β e^{-12N^2\beta} e − 12 N 2 β ν η = γ V ( ⋅ ∣ η ) \nu_\eta=\gamma_V(\cdot\mid\eta) ν η = γ V ( ⋅ ∣ η )
(b) Let π ∗ \pi^* π ∗ Ψ \Psi Ψ x x x V x ∩ S = ∅ V_x\cap S=\varnothing V x ∩ S = ∅ x x x ( σ V x , σ V x ′ ) (\sigma_{V_x},\sigma'_{V_x}) ( σ V x , σ V x ′ ) d V x d_{V_x} d V x μ i \mu_i μ i d V x d_{V_x} d V x ∑ j C V x , j ρ ( j ) \sum_j C_{V_x,j}\rho(j) ∑ j C V x , j ρ ( j ) w ‾ x = max i ∈ V x w i \overline w_x=\max_{i\in V_x}w_i w x = max i ∈ V x w i
∑ i ∈ V x w i ρ ( i ) ≤ w ‾ x ∑ j ∉ V x C V x , j ρ ( j ) , \sum_{i\in V_x}w_i\rho(i)\le\overline w_x\sum_{j\notin V_x}C_{V_x,j}\rho(j), i ∈ V x ∑ w i ρ ( i ) ≤ w x j ∈ / V x ∑ C V x , j ρ ( j ) , and w ‾ x ≤ e λ ρ n w j \overline w_x\le e^{\lambda\rho_n}w_j w x ≤ e λ ρ n w j C V x , j ≠ 0 C_{V_x,j}\ne0 C V x , j = 0 S ′ S' S ′ S S S S S S ρ n \rho_n ρ n S S S ∑ i ∈ S ′ w i ≤ c n e λ ρ n ∑ z ∈ S w z \sum_{i\in S'}w_i\le c_ne^{\lambda\rho_n}\sum_{z\in S}w_z ∑ i ∈ S ′ w i ≤ c n e λ ρ n ∑ z ∈ S w z S ∪ S ′ S\cup S' S ∪ S ′ N N N x x x ∑ x : j ∉ V x C V x , j = K c ( j ) ≤ α N n \sum_{x:j\notin V_x}C_{V_x,j}=K_{c(j)}\le\alpha N_n ∑ x : j ∈ / V x C V x , j = K c ( j ) ≤ α N n
N n ∑ i ∉ S ∪ S ′ w i ρ ( i ) ≤ θ N n [ ∑ i ∉ S ∪ S ′ w i ρ ( i ) + ∑ i ∈ S ′ w i ρ ( i ) ] . N_n\sum_{i\notin S\cup S'}w_i\rho(i)\le\theta N_n\left[\sum_{i\notin S\cup S'}w_i\rho(i)+\sum_{i\in S'}w_i\rho(i)\right]. N n i ∈ / S ∪ S ′ ∑ w i ρ ( i ) ≤ θ N n i ∈ / S ∪ S ′ ∑ w i ρ ( i ) + i ∈ S ′ ∑ w i ρ ( i ) . All sums are finite and ρ ≤ D \rho\le D ρ ≤ D ∑ i ∉ S ∪ S ′ w i ρ ( i ) ≤ θ D 1 − θ ( 1 + c n e λ ρ n ) ∑ z ∈ S w z \sum_{i\notin S\cup S'}w_i\rho(i)\le\frac{\theta D}{1-\theta}(1+c_ne^{\lambda\rho_n})\sum_{z\in S}w_z ∑ i ∈ / S ∪ S ′ w i ρ ( i ) ≤ 1 − θ θ D ( 1 + c n e λ ρ n ) ∑ z ∈ S w z ∑ i ∈ S ′ w i ρ ( i ) ≤ D c n e λ ρ n ∑ z ∈ S w z \sum_{i\in S'}w_i\rho(i)\le Dc_ne^{\lambda\rho_n}\sum_{z\in S}w_z ∑ i ∈ S ′ w i ρ ( i ) ≤ D c n e λ ρ n ∑ z ∈ S w z S = ∅ S=\varnothing S = ∅ S ′ = ∅ S'=\varnothing S ′ = ∅ ρ = 0 \rho=0 ρ = 0
The covariance bound of Corollary 113 follows from part (b) in the same way, with θ D / ( 1 − θ ) \theta D/(1-\theta) θ D / ( 1 − θ ) D ( 1 + c n e λ ρ n ) / ( 1 − θ ) D(1+c_ne^{\lambda\rho_n})/(1-\theta) D ( 1 + c n e λ ρ n ) / ( 1 − θ ) μ σ \mu^\sigma μ σ γ V x \gamma_{V_x} γ V x V x ∩ S H = ∅ V_x\cap S_H=\varnothing V x ∩ S H = ∅
C The plaquette blocks
Table 4 lists the block plaquettes of Sec. V modulo 2 Z 4 2\mathbb{Z}^4 2 Z 4 x x x { μ , ν } \{\mu,\nu\} { μ , ν } x x x { μ , ν } \{\mu,\nu\} { μ , ν } { 0 , 1 } \{0,1\} { 0 , 1 } { 2 , 3 } \{2,3\} { 2 , 3 }
Table 4.
TABLE IV. The sixteen block plaquettes modulo two: the base point x x x x 0 x 1 x 2 x 3 x_0x_1x_2x_3 x 0 x 1 x 2 x 3
In closed form, the plaquette at x x x { 0 , 1 } \{0,1\} { 0 , 1 } x 0 ≡ x 1 x_0\equiv x_1 x 0 ≡ x 1 x x x { 2 , 3 } \{2,3\} { 2 , 3 } x 2 = x 3 x_2=x_3 x 2 = x 3 Z 0 Z_0 Z 0 s = r 2 s=r^2 s = r 2 s ≤ 5 s\leq5 s ≤ 5 ( 5 / 4 ) 40403 / ( 40 ! 41 ! ) (5/4)^{40403}/(40!41!) ( 5/4 ) 40403 / ( 40 ! 41 !) (3) and α \alpha α β W \beta_W β W
For SU(3) the bounds of Lemma 38 are evaluated on the 841 boxes of side 3 / 16 3/16 3/16 r = 3 / 2 r=3/2 r = 3/2 19 / 100 19/100 19/100 r = 38 / 25 r=38/25 r = 38/25 (16) and α 3 \alpha_3 α 3 r = 38 / 25 r=38/25 r = 38/25
The blocks of Lemma 20
Diagonal blocks. Let w a = e i φ w_a=e^{i\varphi} w a = e i φ f p = − cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 f_p=-\cos\phi\,(y_1+y_2-y_3-y_4)^2 f p = − cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 c p = 2 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) c_p=2(y_1y_4-y_1y_2+y_2y_3-y_3y_4) c p = 2 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 )
F = ∑ i α i y i 2 + cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 − 9 5 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) \mathcal{F}=\sum_i\alpha_i y_i^2+\cos\phi\,(y_1+y_2-y_3-y_4)^2-\frac{9}{5}(y_1y_4-y_1y_2+y_2y_3-y_3y_4) F = i ∑ α i y i 2 + cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 − 5 9 ( y 1 y 4 − y 1 y 2 + y 2 y 3 − y 3 y 4 ) with y k = 0 y_k=0 y k = 0 α k ± 1 = 21 10 \alpha_{k\pm1}=\frac{21}{10} α k ± 1 = 10 21 α k + 2 = 5 4 \alpha_{k+2}=\frac{5}{4} α k + 2 = 4 5 cos ϕ \cos\phi cos ϕ cos ϕ = ± 1 \cos\phi=\pm1 cos ϕ = ± 1 k = 1 k=1 k = 1 cos ϕ = 1 \cos\phi=1 cos ϕ = 1 ( y 2 , y 3 , y 4 ) (y_2,y_3,y_4) ( y 2 , y 3 , y 4 )
( 31 10 − 19 10 − 1 − 19 10 9 4 19 10 − 1 19 10 31 10 ) \begin{pmatrix}
\frac{31}{10} & -\frac{19}{10} & -1\\
-\frac{19}{10} & \frac{9}{4} & \frac{19}{10}\\
-1 & \frac{19}{10} & \frac{31}{10}
\end{pmatrix} 10 31 − 10 19 − 1 − 10 19 4 9 10 19 − 1 10 19 10 31 and Table V lists the leading principal minors for every position; all are positive.
Table V. Leading principal minors of the diagonal blocks of Lemma 20.
Pair blocks. Let w a = e i ( χ + ϑ ) w_a=e^{i(\chi+\vartheta)} w a = e i ( χ + ϑ ) w b = e i ( χ − ϑ ) w_b=e^{i(\chi-\vartheta)} w b = e i ( χ − ϑ ) f p f_p f p w a + w b ‾ = 2 cos χ e i ϑ w_a+\overline{w_b}=2\cos\chi\,e^{i\vartheta} w a + w b = 2 cos χ e i ϑ c p c_p c p w a w b ‾ = e 2 i ϑ w_a\overline{w_b}=e^{2i\vartheta} w a w b = e 2 i ϑ M ( m , ϑ ) M(m,\vartheta) M ( m , ϑ ) F \mathcal{F} F m = cos χ m=\cos\chi m = cos χ m m m M ( − 1 , ϑ ) = M ( 1 , ϑ + π ) M(-1,\vartheta)=M(1,\vartheta+\pi) M ( − 1 , ϑ ) = M ( 1 , ϑ + π ) m = 1 m=1 m = 1 e 2 e_2 e 2 ( z 1 , z 3 , z 4 ) (z_1,z_3,z_4) ( z 1 , z 3 , z 4 ) c = cos ϑ c=\cos\vartheta c = cos ϑ
M = ( 21 5 + 2 c − 2 e i ϑ − 9 5 − 2 e i ϑ − 2 e − i ϑ 21 5 + 2 c 9 5 + 2 e − i ϑ − 9 5 − 2 e − i ϑ 9 5 + 2 e i ϑ 5 2 + 2 c ) . M=
\begin{pmatrix}
\frac{21}{5}+2c & -2e^{i\vartheta} & -\frac{9}{5}-2e^{i\vartheta}\\
-2e^{-i\vartheta} & \frac{21}{5}+2c & \frac{9}{5}+2e^{-i\vartheta}\\
-\frac{9}{5}-2e^{-i\vartheta} & \frac{9}{5}+2e^{i\vartheta} & \frac{5}{2}+2c
\end{pmatrix}. M = 5 21 + 2 c − 2 e − i ϑ − 5 9 − 2 e − i ϑ − 2 e i ϑ 5 21 + 2 c 5 9 + 2 e i ϑ − 5 9 − 2 e i ϑ 5 9 + 2 e − i ϑ 2 5 + 2 c . With u = cos 2 ( ϑ / 2 ) ∈ [ 0 , 1 ] u=\cos^2(\vartheta/2)\in[0,1] u = cos 2 ( ϑ /2 ) ∈ [ 0 , 1 ]
20 u + 11 5 , ( 20 u + 1 ) ( 20 u + 21 ) 25 , 400 u ( 40 u 2 − 23 u + 4 ) + 21 250 , \frac{20u+11}{5},\qquad\frac{(20u+1)(20u+21)}{25},\qquad\frac{400u(40u^2-23u+4)+21}{250}, 5 20 u + 11 , 25 ( 20 u + 1 ) ( 20 u + 21 ) , 250 400 u ( 40 u 2 − 23 u + 4 ) + 21 , and 40 u 2 − 23 u + 4 40u^2-23u+4 40 u 2 − 23 u + 4 529 − 640 529-640 529 − 640 20 u + 11 5 \frac{20u+11}{5} 5 20 u + 11 8 u + 1 2 \frac{8u+1}{2} 2 8 u + 1 800 u 2 − 180 u + 53 50 \frac{800u^2-180u+53}{50} 50 800 u 2 − 180 u + 53 32400 − 169600 32400-169600 32400 − 169600
The inequality is not tight. In floating point the least eigenvalue of the whole form over W ∈ U ( N ) W\in U(N) W ∈ U ( N ) N ≤ 4 N\leq4 N ≤ 4 21 10 , 5 4 , 9 10 \frac{21}{10},\frac{5}{4},\frac{9}{10} 10 21 , 4 5 , 10 9 29 2 \frac{29}{2} 2 29
The certified computation
Truncation. For a corner b b b Z Z Z ( b ˉ , b ˉ ) (\bar b,\bar b) ( b ˉ , b ˉ ) E H [ A t ( U D ) B t ( U ˉ D ′ ) ] Tr ( ∑ λ c λ s λ ( U D ) ) \mathbb{E}_{\mathbb{H}}[A_t(UD)B_t(\bar U D')]\operatorname{Tr}(\sum_\lambda c_\lambda s_\lambda(UD)) E H [ A t ( U D ) B t ( U ˉ D ′ )] Tr ( ∑ λ c λ s λ ( U D )) λ 1 − λ 3 ≤ 34 \lambda_1-\lambda_3\leq34 λ 1 − λ 3 ≤ 34 λ 3 ≤ 11 \lambda_3\leq11 λ 3 ≤ 11 B t B_t B t ∣ λ ∣ > 34 |\lambda|>34 ∣ λ ∣ > 34 3 ⋅ 12 > 34 3\cdot12>34 3 ⋅ 12 > 34 T ( ∥ D ∥ ) = ∑ n > 34 ( ∥ D ∥ / 2 ) n / n ! T(\lVert D\rVert)=\sum_{n>34}(\lVert D\rVert/2)^n/n! T (∥ D ∥) = ∑ n > 34 (∥ D ∥ /2 ) n / n ! ∣ D i j − b i j ∣ ≤ ρ |D_{ij}-b_{ij}|\leq\rho ∣ D ij − b ij ∣ ≤ ρ ∥ D ∥ ≤ ∥ b ∥ + 3 ρ \lVert D\rVert\leq\lVert b\rVert+3\rho ∥ D ∥ ≤ ∥ b ∥ + 3 ρ α \alpha α α ! ρ − ∣ α ∣ T ( ∥ b ∥ + 3 ρ ) \alpha!\rho^{-|\alpha|}T(\lVert b\rVert+3\rho) α ! ρ − ∣ α ∣ T (∥ b ∥ + 3 ρ ) 2 − α ∣ e γ / 2 2^{-\alpha}|e^\gamma/2 2 − α ∣ e γ /2 γ = ∑ k ∣ b k ∣ \gamma=\sum_k|b_k| γ = ∑ k ∣ b k ∣ ∣ Tr ( U b ) ∣ ≤ ∑ k ∣ b k ∣ |\operatorname{Tr}(Ub)|\leq\sum_k|b_k| ∣ Tr ( U b ) ∣ ≤ ∑ k ∣ b k ∣ b b b ε α = 2 ∣ α ∣ α ! ρ − ∣ α ∣ T ( ∥ b ∥ + 3 ρ ) \varepsilon_\alpha=2^{|\alpha|}\alpha!\rho^{-|\alpha|}T(\lVert b\rVert+3\rho) ε α = 2 ∣ α ∣ α ! ρ − ∣ α ∣ T (∥ b ∥ + 3 ρ ) ε α e γ / 2 + ( ε α + ε β ) e γ / 2 \varepsilon_\alpha e^{\gamma/2}+(\varepsilon_\alpha+\varepsilon_\beta)e^{\gamma/2} ε α e γ /2 + ( ε α + ε β ) e γ /2 ρ \rho ρ ρ − 6 T \rho^{-6}T ρ − 6 T 1.7 × 10 − 11 1.7\times10^{-11} 1.7 × 1 0 − 11
Weight zero. Conjugation by the diagonal torus T T T Re Tr ( U b ) \operatorname{Re}\operatorname{Tr}(Ub) Re Tr ( U b ) b b b u i j u_{ij} u ij t i t j − 1 t_it_j^{-1} t i t j − 1 T T T Tr ( X ) \operatorname{Tr}(X) Tr ( X ) e 2 ( X ) e_2(X) e 2 ( X ) det X \det X det X X = D D ′ T X=DD'^{\mathsf T} X = D D ′ T det D \det D det D det D ′ \det D' det D ′
Rounding. The truncated series is a Taylor polynomial of total degree six in the 36 real directions of ( D − b , D ′ − b ˉ ) (D-b,D'-\bar b) ( D − b , D ′ − b ˉ ) n n n γ n = n u / ( 1 − n u ) \gamma_n=nu/(1-nu) γ n = n u / ( 1 − n u ) u = 2 − 53 u=2^{-53} u = 2 − 53 [48 ] `.
Separation. The functional c c c 72 × 72 72 \times72 72 × 72 T T T max T ℓ \max_T \ell max T ℓ 512 × 512 512 \times512 512 × 512 1 2 H δ 2 \frac{1}{2}H\delta^2 2 1 H δ 2 δ = 2 π / 512 \delta= 2\pi/512 δ = 2 π /512 H = ∣ c 1 ∣ + ∣ c 2 ∣ + 2 ∣ c 3 ∣ H = |c_1|+|c_2|+2|c_3| H = ∣ c 1 ∣ + ∣ c 2 ∣ + 2∣ c 3 ∣ 10 − 12 10^{-12} 1 0 − 12
The transfer matrix
This appendix recalls the Osterwalder–Schrader reconstruction for the lattice [62 , 63 , 64 ] in the form used for the gaps, following our earlier paper [14 ] .
Let ω \omega ω Z 4 \mathbb{Z}^4 Z 4 Ω \Omega Ω F ↦ Ψ F F \mapsto\Psi_F F ↦ Ψ F T T T 0 ≤ T ≤ 1 0 \le T \le1 0 ≤ T ≤ 1 T Ω = Ω T\Omega= \Omega T Ω = Ω ⟨ Ψ F , T t Ψ H ⟩ = ω ( ϑ ( F ‾ ) τ t H ) \langle\Psi_F,T^t\Psi_H\rangle= \omega(\vartheta(\overline{F})\tau_tH) ⟨ Ψ F , T t Ψ H ⟩ = ω ( ϑ ( F ) τ t H ) ϑ \vartheta ϑ τ t \tau_t τ t 21
Proposition 116. For m > 0 m > 0 m > 0
(a) the spectrum of T T T Ω ⊥ \Omega^\perp Ω ⊥ [ 0 , e − m ] [0,e^{-m}] [ 0 , e − m ]
(b) for every F F F L 2 ( ω ) L^2(\omega) L 2 ( ω ) C F < ∞ C_F < \infty C F < ∞ 0 ≤ ω ( ϑ ( F ‾ ) τ t F ) − ∣ ω ( F ) ∣ 2 ≤ C F e − m t 0 \le\omega(\vartheta(\overline{F})\tau_tF)-|\omega(F)|^2 \le C_Fe^{-mt} 0 ≤ ω ( ϑ ( F ) τ t F ) − ∣ ω ( F ) ∣ 2 ≤ C F e − m t t ≥ 0 t \ge0 t ≥ 0
Under (a) one may take C F = ∥ F ∥ ∞ 2 C_F = \|F\|_\infty^2 C F = ∥ F ∥ ∞ 2 F F F
Proof. If (a) holds, then
ω ( ϑ ( F ‾ ) τ t F ) − ∣ ω ( F ) ∣ 2 = ⟨ Ψ F ′ , T t Ψ F ′ ⟩ \omega(\vartheta(\overline{F})\tau_tF)-|\omega(F)|^2 = \langle\Psi'_F,T^t\Psi'_F\rangle ω ( ϑ ( F ) τ t F ) − ∣ ω ( F ) ∣ 2 = ⟨ Ψ F ′ , T t Ψ F ′ ⟩ with Ψ F ′ = Ψ F − ⟨ Ω , Ψ F ⟩ Ω ∈ Ω ⊥ \Psi'_F = \Psi_F-\langle\Omega,\Psi_F\rangle\Omega\in\Omega^\perp Ψ F ′ = Ψ F − ⟨ Ω , Ψ F ⟩ Ω ∈ Ω ⊥ [ 0 , e − m t ∥ Ψ F ′ ∥ 2 ] [0,e^{-mt}\|\Psi'_F\|^2] [ 0 , e − m t ∥ Ψ F ′ ∥ 2 ] ∥ Ψ F ∥ 2 ≤ ∥ F ∥ ∞ 2 \|\Psi_F\|^2 \le\|F\|_\infty^2 ∥ Ψ F ∥ 2 ≤ ∥ F ∥ ∞ 2 σ \sigma σ T T T Ψ F ′ \Psi'_F Ψ F ′ δ > 0 \delta> 0 δ > 0
σ ( ( e − m + δ , 1 ] ) e ( − m + δ ) t ≤ C F e − m t \sigma((e^{-m+\delta},1])e^{(-m+\delta)t} \le C_Fe^{-mt} σ (( e − m + δ , 1 ]) e ( − m + δ ) t ≤ C F e − m t for all t t t σ ( ( e − m + δ , 1 ] ) = 0 \sigma((e^{-m+\delta},1]) = 0 σ (( e − m + δ , 1 ]) = 0 ( e − m , 1 ] (e^{-m},1] ( e − m , 1 ] Ω ⊥ \Omega^\perp Ω ⊥ Ψ F ′ \Psi'_F Ψ F ′ □ \square □
This appendix collects the facts on spectral measures that the continuum step uses, with proofs.
Lemma 117. Let T T T 0 ≤ T ≤ 1 0 \le T \le1 0 ≤ T ≤ 1 ψ \psi ψ ν \nu ν [ 0 , 1 ] [0,1] [ 0 , 1 ] ⟨ ψ , h ( T ) ψ ⟩ = ∫ h d ν \langle\psi,h(T)\psi\rangle= \int h\,d\nu ⟨ ψ , h ( T ) ψ ⟩ = ∫ h d ν h h h [ 0 , 1 ] [0,1] [ 0 , 1 ] T ψ = ψ T\psi=\psi T ψ = ψ ⟨ ψ ′ , ψ ⟩ = 0 \langle\psi',\psi\rangle=0 ⟨ ψ ′ , ψ ⟩ = 0 ν ( { 1 } ) = 0 \nu(\{1\})=0 ν ({ 1 }) = 0 μ \mu μ ν \nu ν λ ↦ − c − 1 log λ ∈ [ 0 , ∞ ] \lambda\mapsto-c^{-1}\log\lambda\in[0,\infty] λ ↦ − c − 1 log λ ∈ [ 0 , ∞ ] c > 0 c>0 c > 0 ⟨ ψ , T σ / c ψ ⟩ = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) \langle\psi,T^{\sigma/c}\psi\rangle= \int_{[0,\infty]}e^{-\sigma E}\,d\mu(E) ⟨ ψ , T σ / c ψ ⟩ = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) σ > 0 \sigma>0 σ > 0 0 0 0 E = ∞ E=\infty E = ∞
Proof. The spectral theorem [65 Sec. VII.2] gives the measure ν \nu ν T T T [ 0 , 1 ] [0,1] [ 0 , 1 ] ∫ λ n d ν = ⟨ ψ , T n ψ ⟩ \int\lambda^n\,d\nu= \langle\psi,T^n\psi\rangle ∫ λ n d ν = ⟨ ψ , T n ψ ⟩ C [ 0 , 1 ] C[0,1] C [ 0 , 1 ] P = 1 { 1 } ( T ) P=\mathbf{1}_{\{1\}}(T) P = 1 { 1 } ( T ) T T T , s o , so , so λ σ / c = e − σ E \lambda^{\sigma/c}=e^{-\sigma E} λ σ / c = e − σ E λ = e − c E \lambda=e^{-cE} λ = e − c E λ = 0 \lambda=0 λ = 0 E = ∞ E=\infty E = ∞ □ \square □
Lemma 118. Every sequence of probability measures on a compact metric space K K K K K K
Proof. C ( K ) C(K) C ( K ) ( h j ) (h_j) ( h j ) ∫ h j d μ k \int h_j\,d\mu_k ∫ h j d μ k ∥ h j ∥ ∞ \|h_j\|_\infty ∥ h j ∥ ∞ Λ ( h ) = lim ∫ h d μ k \Lambda(h)=\lim\int h\,d\mu_k Λ ( h ) = lim ∫ h d μ k h ∈ C ( K ) h\in C(K) h ∈ C ( K ) Λ \Lambda Λ Λ ( 1 ) = 1 \Lambda(1)=1 Λ ( 1 ) = 1 □ \square □
Lemma 119. Let μ \mu μ [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] ρ ( σ ) = ∫ e − σ E d μ ( E ) \rho(\sigma)=\int e^{-\sigma E}\,d\mu(E) ρ ( σ ) = ∫ e − σ E d μ ( E ) σ > 0 \sigma>0 σ > 0
(a) ρ \rho ρ ρ ( σ ) 2 ≤ ρ ( σ − s ) ρ ( σ + s ) \rho(\sigma)^2 \le\rho(\sigma-s)\rho(\sigma+s) ρ ( σ ) 2 ≤ ρ ( σ − s ) ρ ( σ + s ) 0 < s < σ 0<s<\sigma 0 < s < σ
(b) ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 σ > 0 \sigma>0 σ > 0 μ ( [ 0 , ∞ ) ) > 0 \mu([0,\infty))>0 μ ([ 0 , ∞ )) > 0 ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 σ > 0 \sigma>0 σ > 0
(c) ρ ( σ ) → μ ( [ 0 , ∞ ) ) \rho(\sigma)\to\mu([0,\infty)) ρ ( σ ) → μ ([ 0 , ∞ )) σ ↓ 0 \sigma\downarrow0 σ ↓ 0
(d) If μ \mu μ [ m , ∞ ] [m,\infty] [ m , ∞ ] ρ ( σ ) ≤ e − m σ \rho(\sigma)\le e^{-m\sigma} ρ ( σ ) ≤ e − mσ
(e) For M > 0 M>0 M > 0 μ ( [ 0 , M ] ) ≥ ρ ( σ ) − e − σ M \mu([0,M])\ge\rho(\sigma)-e^{-\sigma M} μ ([ 0 , M ]) ≥ ρ ( σ ) − e − σ M
(f) If μ k → μ \mu_k\to\mu μ k → μ [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] ∫ e − σ E d μ k → ρ ( σ ) \int e^{-\sigma E}\,d\mu_k\to\rho(\sigma) ∫ e − σ E d μ k → ρ ( σ ) σ > 0 \sigma>0 σ > 0
Proof. (a) The integrand decreases in σ \sigma σ
( ∫ e − σ E d μ ) 2 ≤ ∫ e − ( σ − s ) E d μ ∫ e − ( σ + s ) E d μ . \left(\int e^{-\sigma E}\,d\mu\right)^2 \le\int e^{-(\sigma-s)E}\,d\mu\int e^{-(\sigma+s)E}\,d\mu. ( ∫ e − σ E d μ ) 2 ≤ ∫ e − ( σ − s ) E d μ ∫ e − ( σ + s ) E d μ . (b) The integrand is positive exactly on [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) 1 [ 0 , ∞ ) \mathbf{1}_{[0,\infty)} 1 [ 0 , ∞ ) σ ↓ 0 \sigma\downarrow0 σ ↓ 0 e − m σ e^{-m\sigma} e − mσ [ m , ∞ ] [m,\infty] [ m , ∞ ] M M M μ ( [ 0 , M ] ) + e − σ M \mu([0,M])+e^{-\sigma M} μ ([ 0 , M ]) + e − σ M σ > 0 \sigma>0 σ > 0 E ↦ e − σ E E\mapsto e^{-\sigma E} E ↦ e − σ E 0 0 0 ∞ \infty ∞ [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
By (b) and (e), a lower bound ρ ( σ ) ≥ c > 0 \rho(\sigma)\ge c>0 ρ ( σ ) ≥ c > 0 σ \sigma σ c / 2 c/2 c /2 [ 0 , σ − 1 log ( 2 / c ) ] [0,\sigma^{-1}\log(2/c)] [ 0 , σ − 1 log ( 2/ c )] μ k \mu_k μ k k k k [ 0 , ∞ ) [0,\infty) [ 0 , ∞ )
The character expansion
Let G G G d U dU d U ρ λ \rho_\lambda ρ λ d λ d_\lambda d λ χ λ = Tr ρ λ \chi_\lambda=\operatorname{Tr}\rho_\lambda χ λ = Tr ρ λ [22 Chs. 2–4] ).
Lemma 120. Let w ∈ L 1 ( G ) w\in L^1(G) w ∈ L 1 ( G ) w ( U − 1 ) = w ( U ) w(U^{-1})=w(U) w ( U − 1 ) = w ( U )
c ^ λ = d λ − 1 ∫ w χ λ d U . \widehat c_\lambda=d_\lambda^{-1}\int w\chi_\lambda\,dU. c λ = d λ − 1 ∫ w χ λ d U . Then ∫ w ( U ) ρ λ ( U ) d U = c ^ λ I \int w(U)\rho_\lambda(U)\,dU=\widehat c_\lambda I ∫ w ( U ) ρ λ ( U ) d U = c λ I ∫ w ( U ) χ λ ( U − 1 V ) d U = c ^ λ χ λ ( V ) \int w(U)\chi_\lambda(U^{-1}V)\,dU=\widehat c_\lambda\chi_\lambda(V) ∫ w ( U ) χ λ ( U − 1 V ) d U = c λ χ λ ( V )
Proof. M = ∫ w ( U ) ρ λ ( U ) d U M=\int w(U)\rho_\lambda(U)\,dU M = ∫ w ( U ) ρ λ ( U ) d U ρ λ ( V ) \rho_\lambda(V) ρ λ ( V ) w ( V U V − 1 ) = w ( U ) w(VUV^{-1})=w(U) w ( V U V − 1 ) = w ( U ) M = c I M=cI M = c I c = c ^ λ c=\widehat c_\lambda c = c λ ∫ w ( U ) χ λ ( U − 1 V ) d U = Tr ( ∫ w ( U ) ρ λ ( U − 1 ) d U ρ λ ( V ) ) \int w(U)\chi_\lambda(U^{-1}V)\,dU=\operatorname{Tr}\left(\int w(U)\rho_\lambda(U^{-1})\,dU\,\rho_\lambda(V)\right) ∫ w ( U ) χ λ ( U − 1 V ) d U = Tr ( ∫ w ( U ) ρ λ ( U − 1 ) d U ρ λ ( V ) ) U ↦ U − 1 U\mapsto U^{-1} U ↦ U − 1 w ( U − 1 ) = w ( U ) w(U^{-1})=w(U) w ( U − 1 ) = w ( U ) M M M
Lemma 121. Let G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) β > 0 \beta>0 β > 0
w β ( U ) = e β N ReTr U ∫ e β N ReTr V d V . w_\beta(U)=\frac{e^{\frac{\beta}{N}\operatorname{ReTr}U}}{\int e^{\frac{\beta}{N}\operatorname{ReTr}V}\,dV}. w β ( U ) = ∫ e N β ReTr V d V e N β ReTr U . Then c ^ λ ( β ) = d λ − 1 ∫ w β χ λ d U \widehat c_\lambda(\beta)=d_\lambda^{-1}\int w_\beta\chi_\lambda\,dU c λ ( β ) = d λ − 1 ∫ w β χ λ d U 0 ≤ c ^ λ ( β ) ≤ 1 0\le\widehat c_\lambda(\beta)\le1 0 ≤ c λ ( β ) ≤ 1 c ^ λ ( β ) < 1 \widehat c_\lambda(\beta)<1 c λ ( β ) < 1 λ ≠ 0 \lambda\ne0 λ = 0
c ^ λ ( β ) = 1 − N C 2 ( λ ) β + O ( β − 2 ) ( β → ∞ ) , \widehat c_\lambda(\beta)=1-\frac{NC_2(\lambda)}{\beta}+O(\beta^{-2})\qquad(\beta\to\infty), c λ ( β ) = 1 − β N C 2 ( λ ) + O ( β − 2 ) ( β → ∞ ) , where − ∑ a ρ λ ( T a ) 2 = 2 C 2 ( λ ) I -\sum_a\rho_\lambda(T_a)^2=2C_2(\lambda)I − ∑ a ρ λ ( T a ) 2 = 2 C 2 ( λ ) I ( T a ) (T_a) ( T a ) s u ( N ) \mathfrak{su}(N) su ( N ) ∣ X ∣ 2 = − Tr X 2 |X|^2=-\operatorname{Tr}X^2 ∣ X ∣ 2 = − Tr X 2 C 2 = ( N 2 − 1 ) / ( 2 N ) C_2=(N^2-1)/(2N) C 2 = ( N 2 − 1 ) / ( 2 N )
Proof. Since w β ( U ‾ ) = w β ( U ) w_\beta(\overline U)=w_\beta(U) w β ( U ) = w β ( U ) χ λ ( U ‾ ) = χ λ ( U ) ‾ \chi_\lambda(\overline U)=\overline{\chi_\lambda(U)} χ λ ( U ) = χ λ ( U ) ∫ w β χ λ = ∫ w β χ λ ‾ \int w_\beta\chi_\lambda=\int w_\beta\overline{\chi_\lambda} ∫ w β χ λ = ∫ w β χ λ c ^ λ \widehat c_\lambda c λ w β w_\beta w β exp ( β 2 N ( χ F + χ F ‾ ) ) = ∑ m 1 m ! ( β 2 N ) m ( χ F + χ F ‾ ) m \exp\left(\frac{\beta}{2N}(\chi_F+\overline{\chi_F})\right)=\sum_m\frac{1}{m!}\left(\frac{\beta}{2N}\right)^m(\chi_F+\overline{\chi_F})^m exp ( 2 N β ( χ F + χ F ) ) = ∑ m m ! 1 ( 2 N β ) m ( χ F + χ F ) m ∫ w β χ λ ‾ ≥ 0 \int w_\beta\overline{\chi_\lambda}\ge0 ∫ w β χ λ ≥ 0 c ^ λ ≥ 0 \widehat c_\lambda\ge0 c λ ≥ 0 ∣ χ λ ( U ) ∣ ≤ d λ |\chi_\lambda(U)|\le d_\lambda ∣ χ λ ( U ) ∣ ≤ d λ Re χ λ ( U ) = d λ \operatorname{Re}\chi_\lambda(U)=d_\lambda Re χ λ ( U ) = d λ ρ λ ( U ) = I \rho_\lambda(U)=I ρ λ ( U ) = I λ ≠ 0 \lambda\ne0 λ = 0 w β > 0 w_\beta>0 w β > 0 c ^ λ = d λ − 1 ∫ w β Re χ λ < 1 \widehat c_\lambda=d_\lambda^{-1}\int w_\beta\operatorname{Re}\chi_\lambda<1 c λ = d λ − 1 ∫ w β Re χ λ < 1 λ ≠ 0 \lambda\ne0 λ = 0
For the expansion, ReTr U ≤ N \operatorname{ReTr}U\le N ReTr U ≤ N U = I U=I U = I { U = e X : ∣ X ∣ < δ } \{U=e^X:|X|<\delta\} { U = e X : ∣ X ∣ < δ } ∫ e β N ReTr U χ λ \int e^{\frac{\beta}{N}\operatorname{ReTr}U}\chi_\lambda ∫ e N β ReTr U χ λ ∫ e β N ReTr U \int e^{\frac{\beta}{N}\operatorname{ReTr}U} ∫ e N β ReTr U I I I e − c β e^{-c\beta} e − c β I I I U = e X U=e^X U = e X j ( X ) = 1 + O ( ∣ X ∣ 2 ) j(X)=1+O(|X|^2) j ( X ) = 1 + O ( ∣ X ∣ 2 ) X X X X X X Tr X 3 \operatorname{Tr}X^3 Tr X 3
1 N ReTr e X = 1 − 1 2 N ∣ X ∣ 2 + r ( X ) , 0 ≤ r ( X ) ≤ C ∣ X ∣ 4 , \frac{1}{N}\operatorname{ReTr}e^X=1-\frac{1}{2N}|X|^2+r(X),\qquad0\le r(X)\le C|X|^4, N 1 ReTr e X = 1 − 2 N 1 ∣ X ∣ 2 + r ( X ) , 0 ≤ r ( X ) ≤ C ∣ X ∣ 4 , and
d λ − 1 χ λ ( e X ) = 1 + d λ − 1 Tr ρ λ ( X ) + 1 2 d λ Tr ρ λ ( X ) 2 + c 3 ( X ) + O ( ∣ X ∣ 4 ) , d_\lambda^{-1}\chi_\lambda(e^X)=1+d_\lambda^{-1}\operatorname{Tr}\rho_\lambda(X)+\frac{1}{2d_\lambda}\operatorname{Tr}\rho_\lambda(X)^2+c_3(X)+O(|X|^4), d λ − 1 χ λ ( e X ) = 1 + d λ − 1 Tr ρ λ ( X ) + 2 d λ 1 Tr ρ λ ( X ) 2 + c 3 ( X ) + O ( ∣ X ∣ 4 ) , with c 3 c_3 c 3 Tr ρ λ ( X ) = 0 \operatorname{Tr}\rho_\lambda(X)=0 Tr ρ λ ( X ) = 0 s u ( N ) \mathfrak{su}(N) su ( N ) ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ e − β 2 N ∣ X ∣ 2 e^{-\frac{\beta}{2N}|X|^2} e − 2 N β ∣ X ∣ 2 x a x_a x a X = ∑ x a T a X=\sum x_aT_a X = ∑ x a T a N / β N/\beta N / β h = j e β r h=j e^{\beta r} h = j e β r
c ^ λ ( β ) = ⟨ d λ − 1 χ λ ( e X ) h ⟩ ⟨ h ⟩ = 1 + ⟨ 1 2 d λ Tr ρ λ ( X ) 2 h ⟩ ⟨ h ⟩ + ⟨ ( c 3 ( X ) + O ( ∣ X ∣ 4 ) ) h ⟩ ⟨ h ⟩ , \widehat c_\lambda(\beta)=\frac{\left\langle d_\lambda^{-1}\chi_\lambda(e^X)h\right\rangle}{\langle h\rangle}
=\ 1+\frac{\left\langle\frac{1}{2d_\lambda}\operatorname{Tr}\rho_\lambda(X)^2h\right\rangle}{\langle h\rangle}
+\frac{\left\langle\left(c_3(X)+O(|X|^4)\right)h\right\rangle}{\langle h\rangle}, c λ ( β ) = ⟨ h ⟩ ⟨ d λ − 1 χ λ ( e X ) h ⟩ = 1 + ⟨ h ⟩ ⟨ 2 d λ 1 Tr ρ λ ( X ) 2 h ⟩ + ⟨ h ⟩ ⟨ ( c 3 ( X ) + O ( ∣ X ∣ 4 ) ) h ⟩ , the averages being taken over ∣ X ∣ < δ |X|<\delta ∣ X ∣ < δ ⟨ ⋅ ⟩ \langle\cdot\rangle ⟨ ⋅ ⟩ ∣ X ∣ 2 m |X|^{2m} ∣ X ∣ 2 m β − m \beta^{-m} β − m β ∣ r ( X ) ∣ ≤ C β ∣ X ∣ 4 \beta|r(X)|\le C\beta|X|^4 β ∣ r ( X ) ∣ ≤ C β ∣ X ∣ 4 β − 1 \beta^{-1} β − 1 h = 1 + O ( β − 1 ) h=1+O(\beta^{-1}) h = 1 + O ( β − 1 ) h h h X X X Tr X k \operatorname{Tr}X^k Tr X k k k k j j j c 3 c_3 c 3 h − 1 h-1 h − 1 O ( β − 2 ) O(\beta^{-2}) O ( β − 2 ) ⟨ Tr ρ λ ( X ) 2 ⟩ = N β ∑ a Tr ρ λ ( T a ) 2 = − 2 N C 2 ( λ ) d λ β \langle\operatorname{Tr}\rho_\lambda(X)^2\rangle=\frac{N}{\beta}\sum_a\operatorname{Tr}\rho_\lambda(T_a)^2=-\frac{2NC_2(\lambda)d_\lambda}{\beta} ⟨ Tr ρ λ ( X ) 2 ⟩ = β N ∑ a Tr ρ λ ( T a ) 2 = − β 2 N C 2 ( λ ) d λ S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) j j j c ^ j ( β ) = I 2 j + 1 ( β ) / I 1 ( β ) \widehat c_j(\beta)=I_{2j+1}(\beta)/I_1(\beta) c j ( β ) = I 2 j + 1 ( β ) / I 1 ( β ) I n I_n I n C 2 = j ( j + 1 ) C_2=j(j+1) C 2 = j ( j + 1 )
The single-link law of SU(2)
This appendix proves the sharp quarter W 1 ( v b , v b ′ ) ≤ 1 4 ∣ b − b ′ ∣ W_1(v_b,v_{b'})\le\frac14|b-b'| W 1 ( v b , v b ′ ) ≤ 4 1 ∣ b − b ′ ∣ S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) [2 ] and [3 ] use, and the exact value at b = 0 b=0 b = 0 [2 ] uses. The argument is that of our earlier paper [14 ] ; it is given in full so that the present paper is self-contained.
The von Mises–Fisher law
Identify S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) S 3 ⊂ R 4 S^3\subset\mathbb{R}^4 S 3 ⊂ R 4 ReTr U = 2 x 0 ( U ) \operatorname{ReTr}U=2x_0(U) ReTr U = 2 x 0 ( U ) ReTr ( U V ) = 2 ⟨ u , v ⟩ \operatorname{ReTr}(UV)=2\langle u,v\rangle ReTr ( U V ) = 2 ⟨ u , v ⟩ R 4 \mathbb{R}^4 R 4 v ↦ v ˉ v\mapsto\bar v v ↦ v ˉ e e e e e e e e e ReTr ( Q e A p ) \operatorname{ReTr}(Q_eA_p) ReTr ( Q e A p ) A p ∈ S U ( 2 ) A_p\in\mathrm{SU}(2) A p ∈ SU ( 2 ) e e e p p p Q e Q_e Q e
N β ∑ p ∋ e ReTr ( Q e A p ) = ⟨ x , b ⟩ , b = 4 β ∑ p ∋ e A p ‾ ∈ R 4 , N\beta\sum_{p\ni e}\operatorname{ReTr}(Q_eA_p)=\langle x,b\rangle,\qquad b=4\beta\sum_{p\ni e}\overline{A_p}\in\mathbb{R}^4, N β p ∋ e ∑ ReTr ( Q e A p ) = ⟨ x , b ⟩ , b = 4 β p ∋ e ∑ A p ∈ R 4 , with x = Q e ∈ S 3 x=Q_e\in S^3 x = Q e ∈ S 3
κ : = ∣ b ∣ ≤ y : = 24 ∣ β ∣ . (23) \kappa:=|b|\le y:=24|\beta|. \tag*{(23)} κ := ∣ b ∣ ≤ y := 24∣ β ∣. (23) The conditional law of Q e Q_e Q e 22
ν b ( d x ) = Z ( κ ) − 1 e ⟨ x , b ⟩ σ ( d x ) (24) \nu_b(dx)=Z(\kappa)^{-1}e^{\langle x,b\rangle}\sigma(dx) \tag*{(24)} ν b ( d x ) = Z ( κ ) − 1 e ⟨ x , b ⟩ σ ( d x ) (24) on the unit sphere, with σ \sigma σ b = κ e 1 b=\kappa e_1 b = κ e 1 t = x 1 t=x_1 t = x 1 ( 1 − t 2 ) 1 / 2 e κ t (1-t^2)^{1/2}e^{\kappa t} ( 1 − t 2 ) 1/2 e κ t [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] Z ( κ ) = 2 I 1 ( κ ) / κ Z(\kappa)=2I_1(\kappa)/\kappa Z ( κ ) = 2 I 1 ( κ ) / κ I 1 I_1 I 1 Z ( 0 ) = 1 Z(0)=1 Z ( 0 ) = 1
Lemma 122. For every κ ≥ 0 \kappa\geq0 κ ≥ 0 ν b \nu_b ν b S 3 S^3 S 3 Var ν b ( g ) ≤ 1 3 ∫ ∣ ∇ g ∣ 2 d ν b \operatorname{Var}_{\nu_b}(g)\leq\frac{1}{3}\int|\nabla g|^2d\nu_b Var ν b ( g ) ≤ 3 1 ∫ ∣∇ g ∣ 2 d ν b 1 3 \frac{1}{3} 3 1 κ = 0 \kappa=0 κ = 0
Proof. In the polar angle θ ∈ ( 0 , π ) \theta\in(0,\pi) θ ∈ ( 0 , π ) e 1 e_1 e 1 ν b \nu_b ν b L = ∂ θ 2 + b ( θ ) ∂ θ + sin − 2 θ Δ S 2 L=\partial_\theta^2+b(\theta)\partial_\theta+\sin^{-2}\theta\Delta_{S^2} L = ∂ θ 2 + b ( θ ) ∂ θ + sin − 2 θ Δ S 2 b ( θ ) = 2 cot θ − κ sin θ b(\theta)=2\cot\theta-\kappa\sin\theta b ( θ ) = 2 cot θ − κ sin θ ℓ \ell ℓ S 2 S^2 S 2 L ℓ L_\ell L ℓ ℓ ( ℓ + 1 ) sin − 2 θ \ell(\ell+1)\sin^{-2}\theta ℓ ( ℓ + 1 ) sin − 2 θ [9 ] with the trial function ψ = sin θ e a κ cos θ \psi=\sin\theta e^{a\kappa\cos\theta} ψ = sin θ e aκ c o s θ − L 1 ψ ≥ 3 ψ -L_1\psi\geq3\psi − L 1 ψ ≥ 3 ψ a = − 1 5 a=-\frac{1}{5} a = − 5 1 a = − 2 5 a=-\frac{2}{5} a = − 5 2 3 3 3 − L -L − L 3 3 3 κ = 0 \kappa=0 κ = 0 S 3 S^3 S 3 3 3 3
Moments
Let A ( κ ) = E ν b x 1 = Z ′ ( κ ) / Z ( κ ) = I 2 ( κ ) / I 1 ( κ ) A(\kappa)=\mathbb{E}_{\nu_b}x_1=Z'(\kappa)/Z(\kappa)=I_2(\kappa)/I_1(\kappa) A ( κ ) = E ν b x 1 = Z ′ ( κ ) / Z ( κ ) = I 2 ( κ ) / I 1 ( κ )
Lemma 123. For every κ > 0 \kappa>0 κ > 0 b = κ e 1 b=\kappa e_1 b = κ e 1
(i) E x 2 2 = E x 3 2 = E x 4 2 = A / κ \mathbb{E}x_2^2=\mathbb{E}x_3^2=\mathbb{E}x_4^2=A/\kappa E x 2 2 = E x 3 2 = E x 4 2 = A / κ E ( 1 − x 1 2 ) = 3 A / κ \mathbb{E}(1-x_1^2)=3A/\kappa E ( 1 − x 1 2 ) = 3 A / κ x x x diag ( A ′ , A / κ , A / κ , A / κ ) \operatorname{diag}(A',A/\kappa,A/\kappa,A/\kappa) diag ( A ′ , A / κ , A / κ , A / κ ) A ′ = 1 − A 2 − 3 A / κ A'=1-A^2-3A/\kappa A ′ = 1 − A 2 − 3 A / κ
(ii) κ 2 + κ 2 + 4 ≤ A ( κ ) ≤ κ 1 + κ 2 + 9 \frac{\kappa}{2+\sqrt{\kappa^2+4}}\leq A(\kappa)\leq\frac{\kappa}{1+\sqrt{\kappa^2+9}} 2 + κ 2 + 4 κ ≤ A ( κ ) ≤ 1 + κ 2 + 9 κ
(iii) A ′ ≤ A / κ ≤ ( 1 + κ 2 + 9 ) − 1 ≤ 1 4 A'\leq A/\kappa\leq(1+\sqrt{\kappa^2+9})^{-1}\leq\frac14 A ′ ≤ A / κ ≤ ( 1 + κ 2 + 9 ) − 1 ≤ 4 1
(iv) A ( κ ) ≥ κ / ( 3 2 + κ 2 + 25 / 4 ) A(\kappa)\geq\kappa/(\frac32+\sqrt{\kappa^2+25/4}) A ( κ ) ≥ κ / ( 2 3 + κ 2 + 25/4 )
Proof. (i) Integrating by parts against d d t ( 1 − t 2 ) 3 / 2 = − 3 t ( 1 − t 2 ) 1 / 2 \frac{d}{dt}(1-t^2)^{3/2}=-3t(1-t^2)^{1/2} d t d ( 1 − t 2 ) 3/2 = − 3 t ( 1 − t 2 ) 1/2 κ E ( 1 − x 1 2 ) = 3 E x 1 \kappa\mathbb{E}(1-x_1^2)=3\mathbb{E}x_1 κ E ( 1 − x 1 2 ) = 3 E x 1 x 2 , x 3 , x 4 x_2,x_3,x_4 x 2 , x 3 , x 4 ν b \nu_b ν b ∑ i x i 2 = 1 \sum_i x_i^2=1 ∑ i x i 2 = 1 x i ↦ − x i x_i\mapsto-x_i x i ↦ − x i i ≥ 2 i\geq2 i ≥ 2 A ′ = Var x 1 A'=\operatorname{Var}x_1 A ′ = Var x 1 A = ( log Z ) ′ A=(\log Z)' A = ( log Z ) ′ A A A A ′ = 1 − A 2 − 3 A / κ A'=1-A^2-3A/\kappa A ′ = 1 − A 2 − 3 A / κ A ( κ ) = κ / 4 + O ( κ 3 ) A(\kappa)=\kappa/4+O(\kappa^3) A ( κ ) = κ /4 + O ( κ 3 ) 0 0 0
(iii) The middle inequality is (ii). For the first, put s = κ 2 + 4 s=\sqrt{\kappa^2+4} s = κ 2 + 4 4 A / κ ≥ 4 / ( 2 + s ) 4A/\kappa\geq4/(2+s) 4 A / κ ≥ 4/ ( 2 + s ) 1 − A 2 ≤ 1 − κ 2 / ( 2 + s ) 2 1-A^2\leq1-\kappa^2/(2+s)^2 1 − A 2 ≤ 1 − κ 2 / ( 2 + s ) 2
1 − κ 2 ( 2 + s ) 2 ≤ 4 2 + s ⟺ ( 2 + s ) 2 − κ 2 ≤ 4 ( 2 + s ) 1-\frac{\kappa^2}{(2+s)^2}\leq\frac{4}{2+s}
\Longleftrightarrow
(2+s)^2-\kappa^2\leq4(2+s) 1 − ( 2 + s ) 2 κ 2 ≤ 2 + s 4 ⟺ ( 2 + s ) 2 − κ 2 ≤ 4 ( 2 + s ) ⟺ s 2 − κ 2 ≤ 4 , \Longleftrightarrow s^2-\kappa^2\leq4, ⟺ s 2 − κ 2 ≤ 4 , which holds with equality. So A ′ = 1 − A 2 − 3 A / κ ≤ A / κ A'=1-A^2-3A/\kappa\leq A/\kappa A ′ = 1 − A 2 − 3 A / κ ≤ A / κ
The Lipschitz rate of the family
Let δ \delta δ δ ( x , x ′ ) = ∣ x − x ′ ∣ \delta(x,x')=|x-x'| δ ( x , x ′ ) = ∣ x − x ′ ∣ S 3 S^3 S 3 W 1 W_1 W 1 δ \delta δ c ∈ R 4 c\in\mathbb{R}^4 c ∈ R 4 h ∈ R 4 h\in\mathbb{R}^4 h ∈ R 4
L ( c , h ) = sup g Cov c ( g , ⟨ h , x ⟩ ) , (25) L(c,h)=\sup_g\operatorname{Cov}_c(g,\langle h,x\rangle), \tag*{(25)} L ( c , h ) = g sup Cov c ( g , ⟨ h , x ⟩) , (25) the supremum over 1 1 1 g g g δ \delta δ d d t ν c + t h ( g ) = Cov c + t h ( g , ⟨ h , x ⟩ ) \frac{d}{dt}\nu_{c+th}(g)=\operatorname{Cov}_{c+th}(g,\langle h,x\rangle) d t d ν c + t h ( g ) = Cov c + t h ( g , ⟨ h , x ⟩) b , b ′ b,b' b , b ′
W 1 ( ν b , ν b ′ ) ≤ ∣ b − b ′ ∣ sup c ∈ [ b , b ′ ] L ( c , b ′ − b ∣ b ′ − b ∣ ) . (26) W_1(\nu_b,\nu_{b'})\leq|b-b'|\sup_{c\in[b,b']}L\left(c,\frac{b'-b}{|b'-b|}\right). \tag*{(26)} W 1 ( ν b , ν b ′ ) ≤ ∣ b − b ′ ∣ c ∈ [ b , b ′ ] sup L ( c , ∣ b ′ − b ∣ b ′ − b ) . (26) By rotation invariance L ( c , h ) L(c,h) L ( c , h ) κ = ∣ c ∣ \kappa=|c| κ = ∣ c ∣ φ \varphi φ h h h c ⊥ c^\perp c ⊥ L ( κ , φ ) L(\kappa,\varphi) L ( κ , φ )
Lemma 124. For all c c c h h h L ( c , h ) ≤ 1 / ( 2 3 ) L(c,h)\leq1/(2\sqrt{3}) L ( c , h ) ≤ 1/ ( 2 3 )
(i) at c = 0 c=0 c = 0 L ( 0 , h ) = 1 4 L(0,h)=\frac14 L ( 0 , h ) = 4 1 h h h
(ii) if h ⊥ c h\perp c h ⊥ c L ( c , h ) = E ν c x 2 2 = A ( κ ) / κ ≤ 1 4 L(c,h)=\mathbb{E}_{\nu_c}x_2^2=A(\kappa)/\kappa\leq\frac14 L ( c , h ) = E ν c x 2 2 = A ( κ ) / κ ≤ 4 1
(iii) if κ ≥ 88 / 3 ≈ 3.127 \kappa\geq\sqrt{88/3}\approx3.127 κ ≥ 88/3 ≈ 3.127 L ( c , h ) ≤ 1 4 L(c,h)\leq\frac14 L ( c , h ) ≤ 4 1 h h h
Proof. A 1 1 1 δ \delta δ 1 1 1 ∣ ∇ g ∣ ≤ 1 |\nabla g|\leq1 ∣∇ g ∣ ≤ 1 Var g ≤ 1 3 \operatorname{Var}g\leq\frac13 Var g ≤ 3 1 x x x A / κ ≤ 1 4 A/\kappa\leq\frac14 A / κ ≤ 4 1 Var ⟨ h , x ⟩ ≤ A / κ \operatorname{Var}\langle h,x\rangle\leq A/\kappa Var ⟨ h , x ⟩ ≤ A / κ
L ( c , h ) ≤ ( 1 3 A / κ ) 1 / 2 ≤ ( 3 ( 1 + κ 2 + 9 ) ) − 1 / 2 ≤ 1 2 3 . (27) L(c,h)\leq\left(\frac{1}{3}A/\kappa\right)^{1/2}\leq\left(3(1+\sqrt{\kappa^2+9})\right)^{-1/2}\leq\frac{1}{2\sqrt{3}}. \tag*{(27)} L ( c , h ) ≤ ( 3 1 A / κ ) 1/2 ≤ ( 3 ( 1 + κ 2 + 9 ) ) − 1/2 ≤ 2 3 1 . (27) (i) and (ii) are reflection arguments. Let R R R h ⊥ h^\perp h ⊥ ν c \nu_c ν c R R R c = 0 c=0 c = 0 h ⊥ c h\perp c h ⊥ c E ⟨ h , x ⟩ = 0 \mathbb{E}\langle h,x\rangle=0 E ⟨ h , x ⟩ = 0
Cov ( g , ⟨ h , x ⟩ ) = E [ ( g ( x ) − g ( R x ) ) ⟨ h , x ⟩ ; ⟨ h , x ⟩ > 0 ] ≤ E [ 2 ⟨ h , x ⟩ 2 ; ⟨ h , x ⟩ > 0 ] = E ⟨ h , x ⟩ 2 , \operatorname{Cov}(g,\langle h,x\rangle)=\mathbb{E}\left[(g(x)-g(Rx))\langle h,x\rangle;\ \langle h,x\rangle>0\right]
\leq\mathbb{E}\left[2\langle h,x\rangle^2;\ \langle h,x\rangle>0\right]
=\mathbb{E}\langle h,x\rangle^2, Cov ( g , ⟨ h , x ⟩) = E [ ( g ( x ) − g ( R x )) ⟨ h , x ⟩ ; ⟨ h , x ⟩ > 0 ] ≤ E [ 2 ⟨ h , x ⟩ 2 ; ⟨ h , x ⟩ > 0 ] = E ⟨ h , x ⟩ 2 , since δ ( x , R x ) = 2 ∣ ⟨ h , x ⟩ ∣ \delta(x,Rx)=2|\langle h,x\rangle| δ ( x , R x ) = 2∣ ⟨ h , x ⟩ ∣ g = ⟨ h , x ⟩ g=\langle h,x\rangle g = ⟨ h , x ⟩ 1 1 1 c = 0 c=0 c = 0 E ⟨ h , x ⟩ 2 = 1 4 \mathbb{E}\langle h,x\rangle^2=\frac14 E ⟨ h , x ⟩ 2 = 4 1 h ⊥ c h\perp c h ⊥ c E ⟨ h , x ⟩ 2 = A / κ \mathbb{E}\langle h,x\rangle^2=A/\kappa E ⟨ h , x ⟩ 2 = A / κ L ≤ 1 4 L\leq\frac14 L ≤ 4 1 1 + κ 2 + 9 ≥ 16 3 1+\sqrt{\kappa^2+9}\geq\frac{16}{3} 1 + κ 2 + 9 ≥ 3 16 κ 2 ≥ 88 9 \kappa^2\geq\frac{88}{9} κ 2 ≥ 9 88
The reflection settles the directions h ⊥ c h \perp c h ⊥ c c = 0 c=0 c = 0 h h h ν c \nu_c ν c κ < 88 / 3 \kappa<\sqrt{88/3} κ < 88/3
Theorem 125 (The sharp quarter). For every c ∈ R 4 c\in\mathbb{R}^{4} c ∈ R 4 h ∈ R 4 h\in\mathbb{R}^{4} h ∈ R 4
L ( c , h ) ≤ 1 4 , L(c,h)\leq\frac{1}{4}, L ( c , h ) ≤ 4 1 , with equality at c = 0 c=0 c = 0 W 1 ( ν b , ν b ′ ) ≤ 1 4 ∣ b − b ′ ∣ W_{1}(\nu_b,\nu_{b'})\leq\frac{1}{4}\lvert b-b'\rvert W 1 ( ν b , ν b ′ ) ≤ 4 1 ∣ b − b ′ ∣ b , b ′ ∈ R 4 b,b'\in\mathbb{R}^{4} b , b ′ ∈ R 4 1 4 \frac{1}{4} 4 1
The equivalence is (I4) in one direction and the derivative formula above it in the other; the sharpness is Lemma 124(i). The metric matters.23 S 3 S^{3} S 3 B ⊂ R 4 B\subset\mathbb{R}^{4} B ⊂ R 4 B B B 1 4 \frac{1}{4} 4 1 1 4 \frac{1}{4} 4 1
Harmonic extension
Let σ \sigma σ S 3 S^{3} S 3 H k H_k H k k k k S 3 S^{3} S 3 k k k R 4 \mathbb{R}^{4} R 4 H k H_k H k L 2 ( σ ) L^{2}(\sigma) L 2 ( σ ) u k u_k u k u u u H k H_k H k u u u ∫ u d σ = 0 \int u\,\mathrm{d}\sigma=0 ∫ u d σ = 0
Q ( u ) = ∑ k ≥ 1 1 k ∥ u k ∥ L 2 ( σ ) 2 . (28) \mathcal{Q}(u)=\sum_{k\geq1}\frac{1}{k}\lVert u_k\rVert_{L^{2}(\sigma)}^{2}. \tag*{(28)} Q ( u ) = k ≥ 1 ∑ k 1 ∥ u k ∥ L 2 ( σ ) 2 . (28) This is ⟨ u , Λ − 1 u ⟩ \langle u,\Lambda^{-1}u\rangle ⟨ u , Λ − 1 u ⟩ Λ \Lambda Λ B B B H k H_k H k k k k
Lemma 126. Let u u u S 3 S^{3} S 3 ∫ u d σ = 0 \int u\,\mathrm{d}\sigma=0 ∫ u d σ = 0 g g g
∫ S 3 g u d σ ≤ 1 2 Q ( u ) 1 / 2 . \int_{S^{3}}gu\,\mathrm{d}\sigma\leq\frac{1}{2}\mathcal{Q}(u)^{1/2}. ∫ S 3 g u d σ ≤ 2 1 Q ( u ) 1/2 . Proof. The chordal metric is the Euclidean metric of R 4 \mathbb{R}^{4} R 4 S 3 S^{3} S 3 G ( y ) = inf x ∈ S 3 ( g ( x ) + ∣ x − y ∣ ) G(y)=\inf_{x\in S^{3}}(g(x)+\lvert x-y\rvert) G ( y ) = inf x ∈ S 3 ( g ( x ) + ∣ x − y ∣) R 4 \mathbb{R}^{4} R 4 g g g S 3 S^{3} S 3 [59 ] ; by
Rademacher’s theorem ∣ ∇ G ∣ ≤ 1 \lvert\nabla G\rvert\leq1 ∣ ∇ G ∣ ≤ 1 ψ = ∑ k ≥ 1 k − 1 r k u k ( ω ) \psi=\sum_{k\geq1}k^{-1}r^{k}u_k(\omega) ψ = ∑ k ≥ 1 k − 1 r k u k ( ω ) y = r ω y=r\omega y = r ω u u u 24 ∂ r ψ = u \partial_r\psi=u ∂ r ψ = u S 3 S^{3} S 3 G G G ψ \psi ψ ∫ S 3 G ∂ r ψ d S = ∫ B ∇ G ⋅ ∇ ψ d y \int_{S^{3}}G\,\partial_r\psi\,\mathrm{d}S=\int_B\nabla G\cdot\nabla\psi\,\mathrm{d}y ∫ S 3 G ∂ r ψ d S = ∫ B ∇ G ⋅ ∇ ψ d y ∣ S 3 ∣ = 2 π 2 \lvert S^{3}\rvert=2\pi^{2} ∣ S 3 ∣ = 2 π 2 ∣ B ∣ = π 2 / 2 \lvert B\rvert=\pi^{2}/2 ∣ B ∣ = π 2 /2
∫ g u d σ = 1 2 π 2 ∫ B ∇ G ⋅ ∇ ψ d y ≤ ∣ B ∣ 1 / 2 2 π 2 ( ∫ B ∣ ∇ ψ ∣ 2 d y ) 1 / 2 . \int gu\,\mathrm{d}\sigma=\frac{1}{2\pi^{2}}\int_B\nabla G\cdot\nabla\psi\,\mathrm{d}y\leq\frac{\lvert B\rvert^{1/2}}{2\pi^{2}}\left(\int_B\lvert\nabla\psi\rvert^{2}\,\mathrm{d}y\right)^{1/2}. ∫ g u d σ = 2 π 2 1 ∫ B ∇ G ⋅ ∇ ψ d y ≤ 2 π 2 ∣ B ∣ 1/2 ( ∫ B ∣ ∇ ψ ∣ 2 d y ) 1/2 . Green’s formula once more gives ∫ B ∣ ∇ ψ ∣ 2 = ∫ S 3 ψ ∂ r ψ d S = 2 π 2 ∑ k k − 1 ∥ u k ∥ 2 = 2 π 2 Q ( u ) \int_B\lvert\nabla\psi\rvert^{2}=\int_{S^{3}}\psi\,\partial_r\psi\,\mathrm{d}S=2\pi^{2}\sum_k k^{-1}\lVert u_k\rVert^{2}=2\pi^{2}\mathcal{Q}(u) ∫ B ∣ ∇ ψ ∣ 2 = ∫ S 3 ψ ∂ r ψ d S = 2 π 2 ∑ k k − 1 ∥ u k ∥ 2 = 2 π 2 Q ( u ) 1 2 Q ( u ) 1 / 2 \frac{1}{2}\mathcal{Q}(u)^{1/2} 2 1 Q ( u ) 1/2
The lemma is a form of Thomson’s principle. By Kantorovich–Rubinstein duality, sup g ∫ g u d σ \sup_g\int gu\,\mathrm{d}\sigma sup g ∫ g u d σ R 4 \mathbb{R}^{4} R 4 u d σ u\,\mathrm{d}\sigma u d σ 25 ∇ ψ \nabla\psi ∇ ψ c = 0 c=0 c = 0 u = ⟨ h , x ⟩ u=\langle h,x\rangle u = ⟨ h , x ⟩ ψ = ⟨ h , y ⟩ \psi=\langle h,y\rangle ψ = ⟨ h , y ⟩ ∇ ψ = h \nabla\psi=h ∇ ψ = h Q ( u ) = ∥ u ∥ 2 = 1 4 \mathcal{Q}(u)=\lVert u\rVert^{2}=\frac{1}{4} Q ( u ) = ∥ u ∥ 2 = 4 1
Spectral decomposition
Fix c = κ e 1 c=\kappa e_{1} c = κ e 1 κ > 0 \kappa>0 κ > 0 h = cos φ e 1 + sin φ e 2 h=\cos\varphi e_{1}+\sin\varphi e_{2} h = cos φ e 1 + sin φ e 2 ρ = d ν c / d σ = e κ x 1 / Z ( κ ) \rho=\mathrm{d}\nu_c/\mathrm{d}\sigma=e^{\kappa x_{1}}/Z(\kappa) ρ = d ν c / d σ = e κ x 1 / Z ( κ ) Z ( κ ) = ∫ e κ x 1 d σ = 2 I 1 ( κ ) / κ Z(\kappa)=\int e^{\kappa x_{1}}\,\mathrm{d}\sigma=2I_{1}(\kappa)/\kappa Z ( κ ) = ∫ e κ x 1 d σ = 2 I 1 ( κ ) / κ
u = ( ⟨ h , x ⟩ − E ν c ⟨ h , x ⟩ ) ρ , u=(\langle h,x\rangle-\mathbb{E}_{\nu_c}\langle h,x\rangle)\rho, u = (⟨ h , x ⟩ − E ν c ⟨ h , x ⟩) ρ , so that Cov ν c ( g , ⟨ h , x ⟩ ) = ∫ g u d σ \operatorname{Cov}_{\nu_c}(g,\langle h,x\rangle)=\int gu\,\mathrm{d}\sigma Cov ν c ( g , ⟨ h , x ⟩) = ∫ g u d σ ∫ u d σ = 0 \int u\,\mathrm{d}\sigma=0 ∫ u d σ = 0 U n U_n U n
d n ( κ ) = ( n + 1 ) I n + 1 ( κ ) I 1 ( κ ) , n ≥ 0. (29) d_n(\kappa)=\frac{(n+1)I_{n+1}(\kappa)}{I_1(\kappa)},\qquad n\geq0. \tag*{(29)} d n ( κ ) = I 1 ( κ ) ( n + 1 ) I n + 1 ( κ ) , n ≥ 0. (29) Lemma 127. With this notation
Q ( u ) = cos 2 φ Q z ( κ ) + sin 2 φ Q x ( κ ) , \mathcal{Q}(u)=\cos^{2}\varphi\,\mathcal{Q}_z(\kappa)+\sin^{2}\varphi\,\mathcal{Q}_x(\kappa), Q ( u ) = cos 2 φ Q z ( κ ) + sin 2 φ Q x ( κ ) , where
Q z = ∑ n ≥ 1 d n ′ ( κ ) 2 n , Q x = 1 3 κ 2 ∑ n ≥ 1 ( n + 2 ) d n ( κ ) 2 . \mathcal{Q}_z=\sum_{n\geq1}\frac{d_n'(\kappa)^2}{n},\qquad
\mathcal{Q}_x=\frac{1}{3\kappa^{2}}\sum_{n\geq1}(n+2)d_n(\kappa)^2. Q z = n ≥ 1 ∑ n d n ′ ( κ ) 2 , Q x = 3 κ 2 1 n ≥ 1 ∑ ( n + 2 ) d n ( κ ) 2 . Proof. The functions U n ( x 1 ) U_n(x_1) U n ( x 1 ) n ≥ 0 n \ge0 n ≥ 0 U n ( x 1 ) ∈ H n U_n(x_1) \in H_n U n ( x 1 ) ∈ H n L 2 ( σ ) L^2(\sigma) L 2 ( σ ) e κ t = 2 κ ∑ n ( n + 1 ) I n + 1 ( κ ) U n ( t ) e^{\kappa t}=\frac{2}{\kappa}\sum_n(n+1)I_{n+1}(\kappa)U_n(t) e κ t = κ 2 ∑ n ( n + 1 ) I n + 1 ( κ ) U n ( t ) ρ = ∑ n d n ( κ ) U n ( x 1 ) \rho=\sum_n d_n(\kappa)U_n(x_1) ρ = ∑ n d n ( κ ) U n ( x 1 ) d 0 = 1 d_0=1 d 0 = 1
Write u = cos φ u z + sin φ u x u=\cos\varphi\,u_z+\sin\varphi\,u_x u = cos φ u z + sin φ u x u z = ( x 1 − A ) ρ u_z=(x_1-A)\rho u z = ( x 1 − A ) ρ u x = x 2 ρ u_x=x_2\rho u x = x 2 ρ E ν κ c x 1 = A \mathbb{E}_{\nu_\kappa^c}x_1=A E ν κ c x 1 = A E ν κ c x 2 = 0 \mathbb{E}_{\nu_\kappa^c}x_2=0 E ν κ c x 2 = 0 x 2 ↦ − x 2 x_2\mapsto-x_2 x 2 ↦ − x 2 H k H_k H k u z u_z u z u x u_x u x u z u_z u z u x u_x u x H k H_k H k Q ( u ) \mathcal{Q}(u) Q ( u )
Since A = Z ′ / Z A=Z'/Z A = Z ′ / Z u z = ∂ κ ρ = ∑ n d n ′ ( κ ) U n ( x 1 ) u_z=\partial_\kappa\rho=\sum_n d'_n(\kappa)U_n(x_1) u z = ∂ κ ρ = ∑ n d n ′ ( κ ) U n ( x 1 ) u z u_z u z H n H_n H n d n ′ U n ( x 1 ) d'_nU_n(x_1) d n ′ U n ( x 1 ) ∣ d n ′ ∣ \lvert d'_n\rvert ∣ d n ′ ∣ Q z \mathcal{Q}_z Q z
For u x u_x u x y ( t ) = cos t e 1 + sin t e 2 y(t)=\cos t\,e_1+\sin t\,e_2 y ( t ) = cos t e 1 + sin t e 2 ν κ y ( t ) \nu_{\kappa y(t)} ν κ y ( t )
ρ t = e κ ⟨ y ( t ) , x ⟩ Z ( κ ) = ∑ n d n ( κ ) U n ( ⟨ y ( t ) , x ⟩ ) , \rho_t=\frac{e^{\kappa\langle y(t),x\rangle}}{Z(\kappa)}=\sum_n d_n(\kappa)U_n(\langle y(t),x\rangle), ρ t = Z ( κ ) e κ ⟨ y ( t ) , x ⟩ = n ∑ d n ( κ ) U n (⟨ y ( t ) , x ⟩) , and ∂ t ρ t ∣ t = 0 = κ x 2 ρ \left.\partial_t\rho_t\right|_{t=0}=\kappa x_2\rho ∂ t ρ t ∣ t = 0 = κ x 2 ρ U n ( ⟨ y ( t ) , x ⟩ ) U_n(\langle y(t),x\rangle) U n (⟨ y ( t ) , x ⟩) H n H_n H n t t t x 2 U n ′ ( x 1 ) x_2U'_n(x_1) x 2 U n ′ ( x 1 ) t = 0 t=0 t = 0 u x u_x u x H n H_n H n κ − 1 d n x 2 U n ′ ( x 1 ) \kappa^{-1}d_nx_2U'_n(x_1) κ − 1 d n x 2 U n ′ ( x 1 ) n ( n + 2 ) n(n+2) n ( n + 2 ) S 3 S^3 S 3 H n H_n H n f = U n ( x 1 ) f=U_n(x_1) f = U n ( x 1 )
n ( n + 2 ) = n ( n + 2 ) ∥ f ∥ 2 = ∫ ∣ ∇ S 3 f ∣ 2 d σ = ∫ U n ′ ( x 1 ) 2 ( 1 − x 1 2 ) d σ . \begin{aligned}
n(n+2)=n(n+2)\lVert f\rVert^2=\int\lvert\nabla_{S^3}f\rvert^2\,d\sigma\\
&=\int U'_n(x_1)^2(1-x_1^2)\,d\sigma.
\end{aligned} n ( n + 2 ) = n ( n + 2 ) ∥ f ∥ 2 = ∫ ∣ ∇ S 3 f ∣ 2 d σ = ∫ U n ′ ( x 1 ) 2 ( 1 − x 1 2 ) d σ . since the tangential gradient of x 1 x_1 x 1 1 − x 1 2 1-x_1^2 1 − x 1 2 x 1 x_1 x 1 ( x 2 , x 3 , x 4 ) (x_2,x_3,x_4) ( x 2 , x 3 , x 4 ) ( 1 − x 1 2 ) 1 / 2 (1-x_1^2)^{1/2} ( 1 − x 1 2 ) 1/2 E [ x 2 2 ∣ x 1 ] = 1 3 ( 1 − x 1 2 ) \mathbb{E}[x_2^2\mid x_1]=\frac{1}{3}(1-x_1^2) E [ x 2 2 ∣ x 1 ] = 3 1 ( 1 − x 1 2 ) ∥ x 2 U n ′ ( x 1 ) ∥ 2 = 1 3 n ( n + 2 ) \lVert x_2U'_n(x_1)\rVert^2=\frac{1}{3}n(n+2) ∥ x 2 U n ′ ( x 1 ) ∥ 2 = 3 1 n ( n + 2 ) n − 1 κ − 2 d n 2 ⋅ 1 3 n ( n + 2 ) n^{-1}\kappa^{-2}d_n^2\cdot\frac{1}{3}n(n+2) n − 1 κ − 2 d n 2 ⋅ 3 1 n ( n + 2 ) n ≥ 1 n\ge1 n ≥ 1 Q x \mathcal{Q}_x Q x
By Lemmas 126 and 127,
L ( κ , φ ) ≤ 1 2 max ( Q z ( κ ) , Q x ( κ ) ) 1 / 2 , (30) L(\kappa,\varphi)\le\frac{1}{2}\max\bigl(\mathcal{Q}_z(\kappa),\mathcal{Q}_x(\kappa)\bigr)^{1/2}, \tag*{(30)} L ( κ , φ ) ≤ 2 1 max ( Q z ( κ ) , Q x ( κ ) ) 1/2 , (30) and Theorem 125 follows once both parts are at most 1 4 \frac{1}{4} 4 1 1 4 \frac{1}{4} 4 1 κ = 0 \kappa=0 κ = 0 d 1 ∼ κ / 2 d_1\sim\kappa/2 d 1 ∼ κ /2 d 1 ′ → 1 2 d'_1\to\frac{1}{2} d 1 ′ → 2 1 d n = O ( κ n ) d_n=O(\kappa^n) d n = O ( κ n )
Three Bessel sums
The series of Lemma 127 reduce to closed forms. Write I j I_j I j I j ( κ ) I_j(\kappa) I j ( κ )
Lemma 128. For every κ > 0 \kappa>0 κ > 0
∑ j ≥ 1 j 2 I j 2 = κ 4 I 1 ( 2 κ ) , ∑ j ≥ 1 j 3 I j 2 = κ 2 4 ( I 0 2 + I 1 2 ) , \sum_{j\ge1}j^2I_j^2=\frac{\kappa}{4}I_1(2\kappa),\qquad
\sum_{j\ge1}j^3I_j^2=\frac{\kappa^2}{4}(I_0^2+I_1^2), j ≥ 1 ∑ j 2 I j 2 = 4 κ I 1 ( 2 κ ) , j ≥ 1 ∑ j 3 I j 2 = 4 κ 2 ( I 0 2 + I 1 2 ) , I 1 ( 2 κ ) = 2 ∑ j ≥ 1 I j − 1 I j . I_1(2\kappa)=2\sum_{j\ge1}I_{j-1}I_j. I 1 ( 2 κ ) = 2 j ≥ 1 ∑ I j − 1 I j . The first and third identities follow from Parseval’s formula for e κ cos θ = ∑ j ∈ Z I j e i j θ e^{\kappa\cos\theta}=\sum_{j\in\mathbb{Z}}I_je^{ij\theta} e κ c o s θ = ∑ j ∈ Z I j e ij θ I j I_j I j
The transverse part
Proposition 129. Q x ( κ ) ≤ 1 4 \mathcal{Q}_x(\kappa)\le\frac{1}{4} Q x ( κ ) ≤ 4 1 κ ≥ 0 \kappa\ge0 κ ≥ 0
Proof. By (I7), Q x = ( 3 κ 2 I 1 2 ) − 1 ∑ j ≥ 2 j 2 ( j + 1 ) I j 2 \mathcal{Q}_x=(3\kappa^2I_1^2)^{-1}\sum_{j\ge2}j^2(j+1)I_j^2 Q x = ( 3 κ 2 I 1 2 ) − 1 ∑ j ≥ 2 j 2 ( j + 1 ) I j 2 ∑ j ≥ 1 ( j 3 + j 2 ) I j 2 − 2 I 1 2 \sum_{j\ge1}(j^3+j^2)I_j^2-2I_1^2 ∑ j ≥ 1 ( j 3 + j 2 ) I j 2 − 2 I 1 2 Q x ≤ 1 4 \mathcal{Q}_x\le\frac{1}{4} Q x ≤ 4 1
D ( κ ) : = ( 8 + 2 κ 2 ) I 1 2 − κ 2 I 0 2 − κ I 1 ( 2 κ ) ≥ 0. (31) \mathcal{D}(\kappa):=(8+2\kappa^2)I_1^2-\kappa^2I_0^2-\kappa I_1(2\kappa)\ge0. \tag*{(31)} D ( κ ) := ( 8 + 2 κ 2 ) I 1 2 − κ 2 I 0 2 − κ I 1 ( 2 κ ) ≥ 0. (31) Each term is an entire function of κ \kappa κ κ 2 \kappa^2 κ 2 D ( κ ) = ∑ n ≥ 0 D n κ 2 n \mathcal{D}(\kappa)=\sum_{n\ge0}\mathcal{D}_n\kappa^{2n} D ( κ ) = ∑ n ≥ 0 D n κ 2 n D 0 = D 1 = 0 \mathcal{D}_0=\mathcal{D}_1=0 D 0 = D 1 = 0 n = m + 1 ≥ 2 n=m+1\ge2 n = m + 1 ≥ 2
D n = N m 4 m m ! ( m + 1 ) ! ( m + 1 ) ( m + 2 ) , \mathcal{D}_n=\frac{N_m}{4^m m!(m+1)!(m+1)(m+2)}, D n = 4 m m ! ( m + 1 )! ( m + 1 ) ( m + 2 ) N m , with
N m = ( 2 m m ) ( 4 ( 2 m + 1 ) + ( m − 1 ) ( m + 1 ) ( m + 2 ) ) − 4 m ( m + 1 ) ( m + 2 ) . N_m=\binom{2m}{m}\bigl(4(2m+1)+(m-1)(m+1)(m+2)\bigr)-4^m(m+1)(m+2). N m = ( m 2 m ) ( 4 ( 2 m + 1 ) + ( m − 1 ) ( m + 1 ) ( m + 2 ) ) − 4 m ( m + 1 ) ( m + 2 ) . The numerator N m N_m N m m = 0 , 1 , 2 m=0,1,2 m = 0 , 1 , 2 m ≥ 3 m\ge3 m ≥ 3 D \mathcal{D} D D ≥ 0 \mathcal{D}\ge0 D ≥ 0
The first nonzero coefficient is D 4 = 1 2304 \mathcal{D}_4=\frac{1}{2304} D 4 = 2304 1 1 4 − Q x = D / ( 12 κ 2 I 1 2 ) \frac14-\mathcal{Q}_x=\mathcal{D}/(12\kappa^2I_1^2) 4 1 − Q x = D / ( 12 κ 2 I 1 2 ) κ 4 \kappa^4 κ 4 0 0 0 1 4 \frac14 4 1
The zonal part
Let A = I 2 / I 1 A=I_2/I_1 A = I 2 / I 1 d 1 = 2 A d_1=2A d 1 = 2 A Lemma 123 (i), A ′ = 1 − A 2 − 3 A / κ A'=1-A^2-3A/\kappa A ′ = 1 − A 2 − 3 A / κ
Proposition 130. For 0 < κ ≤ 88 / 3 0<\kappa\le\sqrt{88}/3 0 < κ ≤ 88 /3
Q z ( κ ) ≤ 1 4 + 4 A 2 + A 2 4 − A 2 κ − 2 A 2 κ 2 < 1 4 . \mathcal{Q}_z(\kappa)\le\frac14+4A^2+\frac{A^2}{4}-\frac{A}{2\kappa}-\frac{2A^2}{\kappa^2}<\frac14. Q z ( κ ) ≤ 4 1 + 4 A 2 + 4 A 2 − 2 κ A − κ 2 2 A 2 < 4 1 . Proof. The higher modes. With r ν = I ν + 1 / I ν r_\nu=I_{\nu+1}/I_\nu r ν = I ν + 1 / I ν I ν = I ν + 1 + κ ν I ν − 1 I_\nu=I_{\nu+1}+\frac{\kappa}{\nu}I_{\nu-1} I ν = I ν + 1 + ν κ I ν − 1
d n ′ d n = I n + 1 ′ I n + 1 − I 1 ′ I 1 = n κ + r n + 1 − r 1 . \frac{d'_n}{d_n}=\frac{I'_{n+1}}{I_{n+1}}-\frac{I'_1}{I_1}=\frac{n}{\kappa}+r_{n+1}-r_1. d n d n ′ = I n + 1 I n + 1 ′ − I 1 I 1 ′ = κ n + r n + 1 − r 1 . By Lemma 133 of Appendix K, r n + 1 < r 1 r_{n+1}<r_1 r n + 1 < r 1 r 1 = A < κ / 4 r_1=A<\kappa/4 r 1 = A < κ /4 Lemma 123 (iii). Hence − n / κ ≤ n / κ − κ / 4 ≤ d n ′ / d n ≤ n / κ -n/\kappa\le n/\kappa-\kappa/4\le d'_n/d_n\le n/\kappa − n / κ ≤ n / κ − κ /4 ≤ d n ′ / d n ≤ n / κ n ≥ 2 n\ge2 n ≥ 2 κ 2 ≤ 16 \kappa^2\le16 κ 2 ≤ 16 d n ′ 2 ≤ n 2 d n 2 / κ 2 d_n'^2\le n^2d_n^2/\kappa^2 d n ′2 ≤ n 2 d n 2 / κ 2 d 1 ′ = 2 A ′ d'_1=2A' d 1 ′ = 2 A ′
Q z ≤ 4 A ′ 2 + 1 κ 2 ∑ n ≥ 2 n d n 2 = 4 A ′ 2 + S 3 − S 2 − 4 I 1 2 κ 2 I 1 2 , \mathcal{Q}_z\le4A'^2+\frac{1}{\kappa^2}\sum_{n\ge2}n d_n^2=4A'^2+\frac{S_3-S_2-4I_1^2}{\kappa^2I_1^2}, Q z ≤ 4 A ′2 + κ 2 1 n ≥ 2 ∑ n d n 2 = 4 A ′2 + κ 2 I 1 2 S 3 − S 2 − 4 I 1 2 , where S 2 = ∑ j ≥ 1 j 2 I j 2 S_2=\sum_{j\ge1}j^2I_j^2 S 2 = ∑ j ≥ 1 j 2 I j 2 S 3 = ∑ j ≥ 1 j 3 I j 2 S_3=\sum_{j\ge1}j^3I_j^2 S 3 = ∑ j ≥ 1 j 3 I j 2 n d n 2 I 1 2 = ( j − 1 ) I j 2 nd_n^2I_1^2=(j-1)I_j^2 n d n 2 I 1 2 = ( j − 1 ) I j 2 j = n + 1 j=n+1 j = n + 1
The closed form. The recurrences I 0 − I 2 = 2 κ I 1 I_0-I_2=\frac{2}{\kappa}I_1 I 0 − I 2 = κ 2 I 1 I 1 − I 3 = 4 κ I 2 I_1-I_3=\frac{4}{\kappa}I_2 I 1 − I 3 = κ 4 I 2 I 0 / I 1 = 2 / κ + A I_0/I_1=2/\kappa+A I 0 / I 1 = 2/ κ + A I 2 I 3 / I 1 2 = A − 4 A 2 / κ I_2I_3/I_1^2=A-4A^2/\kappa I 2 I 3 / I 1 2 = A − 4 A 2 / κ S 3 / ( κ 2 I 1 2 ) = 1 4 ( 2 / κ + A ) 2 + 1 4 S_3/(\kappa^2 I_1^2)=\frac{1}{4}(2/\kappa+A)^2+\frac{1}{4} S 3 / ( κ 2 I 1 2 ) = 4 1 ( 2/ κ + A ) 2 + 4 1
S 2 κ 2 I 1 2 ≥ I 0 I 1 + I 1 I 2 + I 2 I 3 2 κ I 1 2 = 1 2 κ ( 2 κ + 3 A − 4 A 2 κ ) . \frac{S_2}{\kappa^2 I_1^2}\geq\frac{I_0I_1+I_1I_2+I_2I_3}{2\kappa I_1^2}=\frac{1}{2\kappa}\left(\frac{2}{\kappa}+3A-\frac{4A^2}{\kappa}\right). κ 2 I 1 2 S 2 ≥ 2 κ I 1 2 I 0 I 1 + I 1 I 2 + I 2 I 3 = 2 κ 1 ( κ 2 + 3 A − κ 4 A 2 ) . Collecting terms, Q z ≤ 1 4 + 4 A 2 + T ( A ) Q_z\leq\frac{1}{4}+4A^2+T(A) Q z ≤ 4 1 + 4 A 2 + T ( A ) T ( a ) = a 2 4 − a 2 κ − 2 a 2 κ 2 T(a)=\frac{a^2}{4}-\frac{a}{2\kappa}-\frac{2a^2}{\kappa^2} T ( a ) = 4 a 2 − 2 κ a − κ 2 2 a 2
The barrier. Let s = κ 2 + 25 / 4 s=\sqrt{\kappa^2+25/4} s = κ 2 + 25/4 A ♭ = κ / ( 3 2 + s ) A_\flat=\kappa/(\frac{3}{2}+s) A ♭ = κ / ( 2 3 + s ) Lemma 123 (iv), A ♭ ≤ A ≤ κ / 4 A_\flat\leq A\leq\kappa/4 A ♭ ≤ A ≤ κ /4 a ↦ 1 − a 2 − 3 a / κ a\mapsto1-a^2-3a/\kappa a ↦ 1 − a 2 − 3 a / κ a ≥ 0 a\geq0 a ≥ 0 A ′ ≥ 0 A'\geq0 A ′ ≥ 0 A ′ = Var x 1 A'=\operatorname{Var}x_1 A ′ = Var x 1 0 ≤ A ′ ≤ α : = 1 − A ♭ 2 − 3 A ♭ / κ 0\leq A'\leq\alpha:=1-A_\flat^2-3A_\flat/\kappa 0 ≤ A ′ ≤ α := 1 − A ♭ 2 − 3 A ♭ / κ κ 2 ≤ 12 \kappa^2\leq12 κ 2 ≤ 12 T T T [ 0 , κ / 4 ] [0,\kappa/4] [ 0 , κ /4 ]
T ( a ) − T ( a ′ ) = ( a ′ − a ) 2 κ − ( a + a ′ ) ( κ 2 − 8 ) 4 κ 2 . T(a)-T(a')=(a'-a)\frac{2\kappa-(a+a')(\kappa^2-8)}{4\kappa^2}. T ( a ) − T ( a ′ ) = ( a ′ − a ) 4 κ 2 2 κ − ( a + a ′ ) ( κ 2 − 8 ) . and ( a + a ′ ) ( κ 2 − 8 ) ≤ κ 2 ( κ 2 − 8 ) ≤ 2 κ (a+a')(\kappa^2-8)\leq\frac{\kappa}{2}(\kappa^2-8)\leq2\kappa ( a + a ′ ) ( κ 2 − 8 ) ≤ 2 κ ( κ 2 − 8 ) ≤ 2 κ Q z ≤ 1 4 + 4 α 2 + T ( A ♭ ) Q_z\leq\frac{1}{4}+4\alpha^2+T(A_\flat) Q z ≤ 4 1 + 4 α 2 + T ( A ♭ ) κ 2 = s 2 − 25 4 \kappa^2=s^2-\frac{25}{4} κ 2 = s 2 − 4 25 A ♭ / κ = 2 / ( 2 s + 3 ) A_\flat/\kappa=2/(2s+3) A ♭ / κ = 2/ ( 2 s + 3 ) s s s α = 16 / ( 2 s + 3 ) 2 \alpha=16/(2s+3)^2 α = 16/ ( 2 s + 3 ) 2
4 α 2 + T ( A ♭ ) = ( 2 s − 5 ) p ( s ) 4 ( 2 s + 3 ) 4 , (32) 4\alpha^2+T(A_\flat)=\frac{(2s-5)p(s)}{4(2s+3)^4}, \tag*{(32)} 4 α 2 + T ( A ♭ ) = 4 ( 2 s + 3 ) 4 ( 2 s − 5 ) p ( s ) , (32) p ( s ) = 8 s 3 + 28 s 2 − 98 s − 695. p(s)=8s^3+28s^2-98s-695. p ( s ) = 8 s 3 + 28 s 2 − 98 s − 695. For 0 < κ ≤ 88 / 3 0<\kappa\leq\sqrt{88/3} 0 < κ ≤ 88/3 5 2 < s ≤ 577 / 6 < 4.004 \frac{5}{2}<s\leq\sqrt{577}/6<4.004 2 5 < s ≤ 577 /6 < 4.004 s ≥ 5 2 s\geq\frac{5}{2} s ≥ 2 5 p p p p ′ ( s ) = 24 s 2 + 56 s − 98 > 0 p'(s)=24s^2+56s-98>0 p ′ ( s ) = 24 s 2 + 56 s − 98 > 0 p ( 4.004 ) < − 120 p(4.004)<-120 p ( 4.004 ) < − 120
Proof of Theorem 125
At c = 0 c=0 c = 0 Lemma 124 (i), and for ∣ c ∣ ≥ 88 / 3 |c|\geq\sqrt{88/3} ∣ c ∣ ≥ 88/3 Lemma 124 (iii). For 0 < ∣ c ∣ < 88 / 3 0<|c|<\sqrt{88/3} 0 < ∣ c ∣ < 88/3 Q x , Q z ≤ 1 4 Q_x,Q_z\leq\frac{1}{4} Q x , Q z ≤ 4 1 L ≤ 1 4 L\leq\frac{1}{4} L ≤ 4 1
Computations for the single-link law
Barta's inequality
Let ρ ( θ ) = sin 2 θ e κ cos θ \rho(\theta)=\sin^2\theta e^{\kappa\cos\theta} ρ ( θ ) = sin 2 θ e κ c o s θ ( 0 , π ) (0,\pi) ( 0 , π ) V V V
T = − ρ − 1 ∂ θ ρ ∂ θ + V . T=-\rho^{-1}\partial_\theta\rho\partial_\theta+V. T = − ρ − 1 ∂ θ ρ ∂ θ + V . If ψ > 0 \psi>0 ψ > 0 ( 0 , π ) (0,\pi) ( 0 , π ) ρ ψ 2 = O ( sin 4 θ ) \rho\psi^2=O(\sin^4\theta) ρ ψ 2 = O ( sin 4 θ ) T ψ ≥ λ 0 ψ T\psi\geq\lambda_0\psi T ψ ≥ λ 0 ψ
∫ ( ∣ h ′ ∣ 2 + V h 2 ) ρ d θ ≥ λ 0 ∫ h 2 ρ d θ \int\left(|h'|^2+Vh^2\right)\rho\,d\theta\geq\lambda_0\int h^2\rho\,d\theta ∫ ( ∣ h ′ ∣ 2 + V h 2 ) ρ d θ ≥ λ 0 ∫ h 2 ρ d θ for every smooth h h h 0 0 0 π \pi π h = ψ u h=\psi u h = ψ u ρ ψ ψ ′ u 2 \rho\psi\psi'u^2 ρ ψ ψ ′ u 2
∫ ( ∣ h ′ ∣ 2 + V h 2 ) ρ = ∫ ρ ψ 2 ∣ u ′ ∣ 2 + ∫ T ψ ψ h 2 ρ ≥ λ 0 ∫ h 2 ρ . \int\left(|h'|^2+Vh^2\right)\rho=\int\rho\psi^2|u'|^2+\int\frac{T\psi}{\psi}h^2\rho\geq\lambda_0\int h^2\rho. ∫ ( ∣ h ′ ∣ 2 + V h 2 ) ρ = ∫ ρ ψ 2 ∣ u ′ ∣ 2 + ∫ ψ T ψ h 2 ρ ≥ λ 0 ∫ h 2 ρ . This is the inequality of Barta [9 ] in the form of a ground state transformation.
The computations of Lemma 122
The generator of the reversible diffusion of ν b \nu_b ν b S 3 S^3 S 3 L = Δ S 3 + κ ∇ x 1 ⋅ ∇ L=\Delta_{S^3}+\kappa\nabla x_1\cdot\nabla L = Δ S 3 + κ ∇ x 1 ⋅ ∇ θ \theta θ e 1 e_1 e 1 d θ 2 + sin 2 θ g S 2 d\theta^2+\sin^2\theta\,g_{S^2} d θ 2 + sin 2 θ g S 2 Δ S 3 = ∂ θ 2 + 2 cot θ ∂ θ + sin − 2 θ Δ S 2 \Delta_{S^3}=\partial_\theta^2+2\cot\theta\,\partial_\theta+\sin^{-2}\theta\,\Delta_{S^2} Δ S 3 = ∂ θ 2 + 2 cot θ ∂ θ + sin − 2 θ Δ S 2 ∇ x 1 ⋅ ∇ = − sin θ ∂ θ \nabla x_1\cdot\nabla=-\sin\theta\,\partial_\theta ∇ x 1 ⋅ ∇ = − sin θ ∂ θ h ( θ ) Y ℓ h(\theta)Y_\ell h ( θ ) Y ℓ Y ℓ Y_\ell Y ℓ ℓ \ell ℓ S 2 S^2 S 2 L ℓ = ∂ θ 2 + b ∂ θ − ℓ ( ℓ + 1 ) sin − 2 θ L_\ell=\partial_\theta^2+b\partial_\theta-\ell(\ell+1)\sin^{-2}\theta L ℓ = ∂ θ 2 + b ∂ θ − ℓ ( ℓ + 1 ) sin − 2 θ b = 2 cot θ − κ sin θ = ρ ′ / ρ b=2\cot\theta-\kappa\sin\theta=\rho'/\rho b = 2 cot θ − κ sin θ = ρ ′ / ρ − L ℓ = T -L_\ell=T − L ℓ = T V = ℓ ( ℓ + 1 ) sin − 2 θ V=\ell(\ell+1)\sin^{-2}\theta V = ℓ ( ℓ + 1 ) sin − 2 θ
Degree ℓ ≥ 1 \ell\geq1 ℓ ≥ 1 Take ψ = sin θ e a κ cos θ \psi=\sin\theta\,e^{a\kappa\cos\theta} ψ = sin θ e aκ c o s θ ψ ′ / ψ = cot θ − a κ sin θ \psi'/\psi=\cot\theta-a\kappa\sin\theta ψ ′ / ψ = cot θ − aκ sin θ ψ ′ ′ / ψ = ( ψ ′ / ψ ) 2 + ( ψ ′ / ψ ) ′ \psi''/\psi=(\psi'/\psi)^2+(\psi'/\psi)' ψ ′′ / ψ = ( ψ ′ / ψ ) 2 + ( ψ ′ / ψ ) ′
− L 1 ψ ψ = 3 + ( 1 + 5 a ) κ cos θ − a ( 1 + a ) κ 2 sin 2 θ . -\frac{L_1\psi}{\psi}=3+(1+5a)\kappa\cos\theta-a(1+a)\kappa^2\sin^2\theta. − ψ L 1 ψ = 3 + ( 1 + 5 a ) κ cos θ − a ( 1 + a ) κ 2 sin 2 θ . For a = − 1 5 a=-\frac{1}{5} a = − 5 1 3 + 4 25 κ 2 sin 2 θ ≥ 3 3+\frac{4}{25}\kappa^2\sin^2\theta\geq3 3 + 25 4 κ 2 sin 2 θ ≥ 3 V = ℓ ( ℓ + 1 ) sin − 2 θ ≥ 2 sin − 2 θ V=\ell(\ell+1)\sin^{-2}\theta\geq2\sin^{-2}\theta V = ℓ ( ℓ + 1 ) sin − 2 θ ≥ 2 sin − 2 θ ℓ ≥ 1 \ell\geq1 ℓ ≥ 1 − L ℓ ≥ 3 -L_\ell\geq3 − L ℓ ≥ 3 ℓ ≥ 1 \ell\geq1 ℓ ≥ 1
Degree ℓ = 0 \ell=0 ℓ = 0 Let g ( θ ) g(\theta) g ( θ ) − L 0 g = λ g -L_0g=\lambda g − L 0 g = λ g g g g g ′ ( 0 ) = g ′ ( π ) = 0 g'(0)=g'(\pi)=0 g ′ ( 0 ) = g ′ ( π ) = 0 g ′ ′ + b g ′ = − λ g g''+bg'=-\lambda g g ′′ + b g ′ = − λ g h = g ′ h=g' h = g ′
− ( h ′ ′ + b h ′ ) − b ′ h = λ h , − b ′ = 2 sin 2 θ + κ cos θ , -(h''+bh')-b'h=\lambda h,\qquad-b'=\frac{2}{\sin^2\theta}+\kappa\cos\theta, − ( h ′′ + b h ′ ) − b ′ h = λh , − b ′ = sin 2 θ 2 + κ cos θ , so h h h T T T V = 2 sin − 2 θ + κ cos θ V=2\sin^{-2}\theta+\kappa\cos\theta V = 2 sin − 2 θ + κ cos θ λ \lambda λ h ≠ 0 h\neq0 h = 0 ψ \psi ψ
T ψ ψ = 3 + ( 2 + 5 a ) κ cos θ − a ( 1 + a ) κ 2 sin 2 θ , \frac{T\psi}{\psi}=3+(2+5a)\kappa\cos\theta-a(1+a)\kappa^2\sin^2\theta, ψ T ψ = 3 + ( 2 + 5 a ) κ cos θ − a ( 1 + a ) κ 2 sin 2 θ , which for a = − 2 5 a=-\frac{2}{5} a = − 5 2 3 + 6 25 κ 2 sin 2 θ ≥ 3 3+\frac{6}{25}\kappa^2\sin^2\theta\geq3 3 + 25 6 κ 2 sin 2 θ ≥ 3 h h h λ ≥ 3 \lambda\geq3 λ ≥ 3 ℓ ≥ 1 \ell\geq1 ℓ ≥ 1 − L -L − L 3 3 3 Lemma 122 ; the spectrum is discrete because S 3 S^3 S 3
Moments
The density of t = x 1 t=x_1 t = x 1 ν b \nu_b ν b ( 1 − t 2 ) 1 / 2 e κ t (1-t^2)^{1/2}e^{\kappa t} ( 1 − t 2 ) 1/2 e κ t ϕ \phi ϕ [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
d d t [ ( 1 − t 2 ) 3 / 2 e κ t ] = ( 1 − t 2 ) 1 / 2 e κ t ( κ ( 1 − t 2 ) − 3 t ) , \frac{d}{dt}\left[(1-t^2)^{3/2}e^{\kappa t}\right]=(1-t^2)^{1/2}e^{\kappa t}\left(\kappa(1-t^2)-3t\right), d t d [ ( 1 − t 2 ) 3/2 e κ t ] = ( 1 − t 2 ) 1/2 e κ t ( κ ( 1 − t 2 ) − 3 t ) , and integration by parts gives E [ ϕ ′ ( t ) ( 1 − t 2 ) ] = E [ ϕ ( t ) ( 3 t − κ ( 1 − t 2 ) ) ] \mathbb{E}[\phi'(t)(1-t^2)]=\mathbb{E}[\phi(t)(3t-\kappa(1-t^2))] E [ ϕ ′ ( t ) ( 1 − t 2 )] = E [ ϕ ( t ) ( 3 t − κ ( 1 − t 2 ))] ϕ = 1 \phi=1 ϕ = 1 E ( t 2 − 1 ) = − 2 3 E t \mathbb{E}(t^2-1)=-\frac{2}{3}\mathbb{E}t E ( t 2 − 1 ) = − 3 2 E t Lemma 123 (i). With ϕ = t k \phi=t^k ϕ = t k
m k + 2 = m k − k + 3 κ m k + 1 + k κ m k − 1 , m k = E X 1 k , m_{k+2}=m_k-\frac{k+3}{\kappa}m_{k+1}+\frac{k}{\kappa}m_{k-1},\qquad m_k=\mathbb{E}X_1^k, m k + 2 = m k − κ k + 3 m k + 1 + κ k m k − 1 , m k = E X 1 k , from m 0 = 1 m_0=1 m 0 = 1 m 1 = A m_1=A m 1 = A ν b \nu_b ν b κ \kappa κ A ( κ ) A(\kappa) A ( κ ) 26
The Riccati barriers
By Lemma 123 (i), A A A A ′ = F ( κ , A ) : = 1 − A 2 − 3 A / κ A'=\mathcal{F}(\kappa,A):=1-A^2-3A/\kappa A ′ = F ( κ , A ) := 1 − A 2 − 3 A / κ ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) A ( κ ) = κ 4 − κ 3 96 + O ( κ 5 ) A(\kappa)=\frac{\kappa}{4}-\frac{\kappa^3}{96}+O(\kappa^5) A ( κ ) = 4 κ − 96 κ 3 + O ( κ 5 ) I 1 I_1 I 1 I 2 I_2 I 2 s = κ 2 + 4 s=\sqrt{\kappa^2+4} s = κ 2 + 4 u = κ 2 + 9 u=\sqrt{\kappa^2+9} u = κ 2 + 9 ϕ − = κ / ( 2 + s ) \phi_-=\kappa/(2+s) ϕ − = κ / ( 2 + s ) ϕ + = κ / ( 1 + u ) \phi_+=\kappa/(1+u) ϕ + = κ / ( 1 + u )
ϕ − ′ − F ( κ , ϕ − ) = − κ 2 s ( s + 2 ) 2 ≤ 0 , \phi_-' - \mathcal{F}(\kappa,\phi_-)=-\frac{\kappa^2}{s(s+2)^2}\leq0, ϕ − ′ − F ( κ , ϕ − ) = − s ( s + 2 ) 2 κ 2 ≤ 0 , ϕ + ′ − F ( κ , ϕ + ) = ( u − 3 ) 2 u ( 1 + u ) 2 ≥ 0 , \phi_+'-\mathcal{F}(\kappa,\phi_+)=\frac{(u-3)^2}{u(1+u)^2}\geq0, ϕ + ′ − F ( κ , ϕ + ) = u ( 1 + u ) 2 ( u − 3 ) 2 ≥ 0 , and ϕ ± = κ 4 + O ( κ 3 ) \phi_\pm=\frac{\kappa}{4}+O(\kappa^3) ϕ ± = 4 κ + O ( κ 3 ) 27 3 A / κ 3A/\kappa 3 A / κ D = A − ϕ − D=A-\phi_- D = A − ϕ −
D ′ = F ( κ , A ) − F ( κ , ϕ − ) − ( ϕ − ′ − F ( κ , ϕ − ) ) = − ( A + ϕ − + 3 κ ) D + R , \begin{aligned}
D'=\mathcal{F}(\kappa,A)-\mathcal{F}(\kappa,\phi_-)-\left(\phi_-' -\mathcal{F}(\kappa,\phi_-)\right) \\
&=-\left(A+\phi_-+\frac{3}{\kappa}\right)D+R,
\end{aligned} D ′ = F ( κ , A ) − F ( κ , ϕ − ) − ( ϕ − ′ − F ( κ , ϕ − ) ) = − ( A + ϕ − + κ 3 ) D + R , with R = κ 2 / ( s ( s + 2 ) 2 ) ≥ 0 R=\kappa^2/(s(s+2)^2)\geq0 R = κ 2 / ( s ( s + 2 ) 2 ) ≥ 0 μ ( κ ) = κ 3 exp ∫ 0 κ ( A + ϕ − ) \mu(\kappa)=\kappa^3\exp\int_0^\kappa(A+\phi_-) μ ( κ ) = κ 3 exp ∫ 0 κ ( A + ϕ − ) ( μ D ) ′ = μ R ≥ 0 (\mu D)'=\mu R\geq0 ( μ D ) ′ = μ R ≥ 0 μ D = O ( κ 6 ) → 0 \mu D=O(\kappa^6)\to0 μ D = O ( κ 6 ) → 0 κ → 0 \kappa\to0 κ → 0 μ D ≥ 0 \mu D\geq0 μ D ≥ 0 A ≥ ϕ − A\geq\phi_- A ≥ ϕ − D = ϕ + − A D=\phi_+-A D = ϕ + − A R = ( u − 3 ) 2 / ( u ( 1 + u ) 2 ) R=(u-3)^2/(u(1+u)^2) R = ( u − 3 ) 2 / ( u ( 1 + u ) 2 ) A ≤ ϕ + A\leq\phi_+ A ≤ ϕ + Lemma 123 (ii).
For (iv) let s = κ 2 + 25 / 4 s=\sqrt{\kappa^2+25/4} s = κ 2 + 25/4 ϕ ♭ = κ / ( 3 / 2 + s ) \phi_\flat=\kappa/(3/2+s) ϕ ♭ = κ / ( 3/2 + s ) s 2 − 9 / 4 = κ 2 + 4 s^2-9/4=\kappa^2+4 s 2 − 9/4 = κ 2 + 4
ϕ ♭ ′ − F ( κ , ϕ ♭ ) = − 5 ( s − 5 / 2 ) 2 s ( 3 / 2 + s ) 2 ≤ 0 , \phi_\flat'-\mathcal{F}(\kappa,\phi_\flat)=-\frac{5(s-5/2)}{2s(3/2+s)^2}\leq0, ϕ ♭ ′ − F ( κ , ϕ ♭ ) = − 2 s ( 3/2 + s ) 2 5 ( s − 5/2 ) ≤ 0 , so ϕ ♭ \phi_\flat ϕ ♭ ϕ ♭ = κ 4 − κ 3 80 + O ( κ 5 ) \phi_\flat=\frac{\kappa}{4}-\frac{\kappa^3}{80}+O(\kappa^5) ϕ ♭ = 4 κ − 80 κ 3 + O ( κ 5 ) 28 A − ϕ ♭ = κ 3 480 + O ( κ 5 ) A-\phi_\flat=\frac{\kappa^3}{480}+O(\kappa^5) A − ϕ ♭ = 480 κ 3 + O ( κ 5 ) D = A − ϕ ♭ D=A-\phi_\flat D = A − ϕ ♭ R = 5 ( s − 5 / 2 ) / ( 2 s ( 3 / 2 + s ) 2 ) ≥ 0 R=5(s-5/2)/(2s(3/2+s)^2)\geq0 R = 5 ( s − 5/2 ) / ( 2 s ( 3/2 + s ) 2 ) ≥ 0 μ = κ 3 exp ∫ 0 κ ( A + ϕ ♭ ) \mu=\kappa^3\exp\int_0^\kappa(A+\phi_\flat) μ = κ 3 exp ∫ 0 κ ( A + ϕ ♭ ) μ D = O ( κ 6 ) \mu D=O(\kappa^6) μ D = O ( κ 6 ) ( μ D ) ′ = μ R ≥ 0 (\mu D)'=\mu R\geq0 ( μ D ) ′ = μ R ≥ 0 A ≥ ϕ ♭ A\geq\phi_\flat A ≥ ϕ ♭ 3 / 2 + κ 2 + 25 / 4 < 2 + κ 2 + 4 3/2+\sqrt{\kappa^2+25/4}<2+\sqrt{\kappa^2+4} 3/2 + κ 2 + 25/4 < 2 + κ 2 + 4 κ > 0 \kappa>0 κ > 0
K Bessel functions
This appendix proves the auxiliary statements used in Appendix I for Theorem 125 . Throughout, I j = I j ( κ ) I_j=I_j(\kappa) I j = I j ( κ ) κ > 0 \kappa>0 κ > 0 I − j = I j I_{-j}=I_j I − j = I j j j j
I ν ( κ ) = ∑ i ≥ 0 ( κ / 2 ) 2 i + ν i ! ( i + ν ) ! , I_\nu(\kappa)=\sum_{i\geq0}\frac{(\kappa/2)^{2i+\nu}}{i!(i+\nu)!}, I ν ( κ ) = i ≥ 0 ∑ i ! ( i + ν )! ( κ /2 ) 2 i + ν , the integral I j ( κ ) = π − 1 ∫ 0 π e κ cos θ cos j θ d θ I_j(\kappa)=\pi^{-1}\int_0^\pi e^{\kappa\cos\theta}\cos j\theta\,d\theta I j ( κ ) = π − 1 ∫ 0 π e κ c o s θ cos j θ d θ e κ cos θ = ∑ j ∈ Z I j e i j θ e^{\kappa\cos\theta}=\sum_{j\in\mathbb{Z}}I_je^{ij\theta} e κ c o s θ = ∑ j ∈ Z I j e ij θ
I j − 1 − I j + 1 = 2 j κ I j , I j ′ = I j + 1 + j κ I j , (33) I_{j-1}-I_{j+1}=\frac{2j}{\kappa}I_j,\qquad I_j'=I_{j+1}+\frac{j}{\kappa}I_j, \tag*{(33)} I j − 1 − I j + 1 = κ 2 j I j , I j ′ = I j + 1 + κ j I j , (33) all of which are in Watson’s treatise [76 ] , §§3.7, 3.71, 2.22. All series below converge absolutely, because 0 < I j ≤ ( κ / 2 ) j e κ / j ! 0<I_j\leq(\kappa/2)^j e^\kappa/j! 0 < I j ≤ ( κ /2 ) j e κ / j !
Zonal harmonics
Under σ \sigma σ x 1 x_1 x 1 2 π ( 1 − t 2 ) 1 / 2 \frac{2}{\pi}(1-t^2)^{1/2} π 2 ( 1 − t 2 ) 1/2 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] { x 1 = t } \{x_1=t\} { x 1 = t } 1 − t 2 1-t^2 1 − t 2 ( 1 − t 2 ) − 1 / 2 (1-t^2)^{-1/2} ( 1 − t 2 ) − 1/2 U n ( cos θ ) = sin ( ( n + 1 ) θ ) / sin θ U_n(\cos\theta)=\sin((n+1)\theta)/\sin\theta U n ( cos θ ) = sin (( n + 1 ) θ ) / sin θ
∫ U n U m d σ = 2 π ∫ 0 π sin ( ( n + 1 ) θ ) sin ( ( m + 1 ) θ ) d θ = δ n m , \int U_nU_m\,d\sigma=\frac{2}{\pi}\int_0^\pi\sin((n+1)\theta)\sin((m+1)\theta)\,d\theta=\delta_{nm}, ∫ U n U m d σ = π 2 ∫ 0 π sin (( n + 1 ) θ ) sin (( m + 1 ) θ ) d θ = δ nm , and U n = C n ( 1 ) U_n=C_n^{(1)} U n = C n ( 1 ) ( 4 − 2 ) / 2 = 1 (4-2)/2=1 ( 4 − 2 ) /2 = 1 U n ( ⟨ y , x ⟩ ) U_n(\langle y,x\rangle) U n (⟨ y , x ⟩) n n n S 3 S^3 S 3 y y y ∣ x ∣ n U n ( ⟨ y , x ⟩ / ∣ x ∣ ) |x|^nU_n(\langle y,x\rangle/|x|) ∣ x ∣ n U n (⟨ y , x ⟩ /∣ x ∣ ) R 4 \mathbb{R}^4 R 4 [72 ] , Ch. IV, §2.
Lemma 131. e κ t = 2 κ ∑ n ≥ 0 ( n + 1 ) I n + 1 ( κ ) U n ( t ) e^{\kappa t}=\frac{2}{\kappa}\sum_{n\geq0}(n+1)I_{n+1}(\kappa)U_n(t) e κ t = κ 2 ∑ n ≥ 0 ( n + 1 ) I n + 1 ( κ ) U n ( t ) L 2 ( ( π / 2 ) ( 1 − t 2 ) 1 / 2 d t ) L^2((\pi/2)(1-t^2)^{1/2}dt) L 2 (( π /2 ) ( 1 − t 2 ) 1/2 d t ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ]
Proof. With t = cos θ t=\cos\theta t = cos θ 2 sin θ sin ( ( n + 1 ) θ ) = cos n θ − cos ( n + 2 ) θ 2\sin\theta\sin((n+1)\theta)=\cos n\theta-\cos(n+2)\theta 2 sin θ sin (( n + 1 ) θ ) = cos n θ − cos ( n + 2 ) θ n n n
2 π ∫ 0 π e κ cos θ sin ( ( n + 1 ) θ ) sin θ d θ = I n − I n + 2 = 2 ( n + 1 ) κ I n + 1 , \begin{aligned}
\frac{2}{\pi}\int_0^\pi e^{\kappa\cos\theta}\sin((n+1)\theta)\sin\theta\,d\theta=I_n-I_{n+2} \\
&=\frac{2(n+1)}{\kappa}I_{n+1},
\end{aligned} π 2 ∫ 0 π e κ c o s θ sin (( n + 1 ) θ ) sin θ d θ = I n − I n + 2 = κ 2 ( n + 1 ) I n + 1 , by (K1). Since ∣ U n ∣ ≤ n + 1 |U_n|\leq n+1 ∣ U n ∣ ≤ n + 1 I n + 1 I_{n+1} I n + 1
Proof of Lemma 128
Let F ( θ ) = e κ cos θ = ∑ j I j e i j θ F(\theta)=e^{\kappa\cos\theta}=\sum_jI_je^{ij\theta} F ( θ ) = e κ c o s θ = ∑ j I j e ij θ
F ′ ( θ ) = − κ sin θ F ( θ ) = ∑ j i j I j e i j θ , F'(\theta)=-\kappa\sin\theta F(\theta)=\sum_jijI_je^{ij\theta}, F ′ ( θ ) = − κ sin θ F ( θ ) = j ∑ ij I j e ij θ , and Parseval’s formula gives
∑ j ∈ Z j 2 I j 2 = κ 2 2 π ∫ 0 2 π sin 2 θ e 2 κ cos θ d θ = κ 2 2 ( I 0 ( 2 κ ) − I 2 ( 2 κ ) ) = κ 2 I 1 ( 2 κ ) , (34) \begin{aligned}
\sum_{j\in\mathbb{Z}}j^2I_j^2=\frac{\kappa^2}{2\pi}\int_0^{2\pi}\sin^2\theta\,e^{2\kappa\cos\theta}\,d\theta\\
&=\frac{\kappa^2}{2}\left(I_0(2\kappa)-I_2(2\kappa)\right)=\frac{\kappa}{2}I_1(2\kappa), \tag*{(34)}
\end{aligned} j ∈ Z ∑ j 2 I j 2 = 2 π κ 2 ∫ 0 2 π sin 2 θ e 2 κ c o s θ d θ = 2 κ 2 ( I 0 ( 2 κ ) − I 2 ( 2 κ ) ) = 2 κ I 1 ( 2 κ ) , (34) the last step by (K1) at the argument 2 κ 2\kappa 2 κ e i θ e^{i\theta} e i θ F 2 = e 2 κ cos θ F_2=e^{2\kappa\cos\theta} F 2 = e 2 κ c o s θ I 1 ( 2 κ ) = ∑ j ∈ Z I j I 1 − j I_1(2\kappa)=\sum_{j\in\mathbb{Z}}I_jI_{1-j} I 1 ( 2 κ ) = ∑ j ∈ Z I j I 1 − j j ≤ 0 j\leq0 j ≤ 0 j ≥ 1 j\geq1 j ≥ 1
For the second, j I j = κ 2 ( I j − 1 − I j + 1 ) jI_j=\frac{\kappa}{2}(I_{j-1}-I_{j+1}) j I j = 2 κ ( I j − 1 − I j + 1 )
∑ j ≥ 1 j 3 I j 2 = κ 2 4 ∑ j ≥ 1 j ( I j − 1 2 + I j + 1 2 − 2 I j − 1 I j + 1 ) . \sum_{j\geq1}j^3I_j^2=\frac{\kappa^2}{4}\sum_{j\geq1}j\left(I_{j-1}^2+I_{j+1}^2-2I_{j-1}I_{j+1}\right). j ≥ 1 ∑ j 3 I j 2 = 4 κ 2 j ≥ 1 ∑ j ( I j − 1 2 + I j + 1 2 − 2 I j − 1 I j + 1 ) . Shifting indices, ∑ j ≥ 1 j ( I j − 1 2 + I j + 1 2 ) = I 0 2 + ∑ m ≥ 1 2 m I m 2 \sum_{j\geq1}j(I_{j-1}^2+I_{j+1}^2)=I_0^2+\sum_{m\geq1}2mI_m^2 ∑ j ≥ 1 j ( I j − 1 2 + I j + 1 2 ) = I 0 2 + ∑ m ≥ 1 2 m I m 2 ∑ m ≥ 1 2 m I m 2 = κ ∑ m ≥ 1 ( I m − 1 I m − I m I m + 1 ) = κ I 0 I 1 \sum_{m\geq1}2mI_m^2=\kappa\sum_{m\geq1}(I_{m-1}I_m-I_mI_{m+1})=\kappa I_0I_1 ∑ m ≥ 1 2 m I m 2 = κ ∑ m ≥ 1 ( I m − 1 I m − I m I m + 1 ) = κ I 0 I 1 j = 1 2 ( ( j − 1 ) + ( j + 1 ) ) j=\frac{1}{2}((j-1)+(j+1)) j = 2 1 (( j − 1 ) + ( j + 1 )) ( j − 1 ) I j − 1 (j-1)I_{j-1} ( j − 1 ) I j − 1 ( j + 1 ) I j + 1 (j+1)I_{j+1} ( j + 1 ) I j + 1 I − 1 = I 1 I_{-1}=I_1 I − 1 = I 1 j = 1 j=1 j = 1
j I j − 1 I j + 1 = κ 4 [ ( I j − 1 I j − I j I j + 1 ) + ( I j − 2 I j + 1 − I j − 1 I j + 2 ) ] . jI_{j-1}I_{j+1}=\frac{\kappa}{4}\left[(I_{j-1}I_j-I_jI_{j+1})+(I_{j-2}I_{j+1}-I_{j-1}I_{j+2})\right]. j I j − 1 I j + 1 = 4 κ [ ( I j − 1 I j − I j I j + 1 ) + ( I j − 2 I j + 1 − I j − 1 I j + 2 ) ] . Both brackets telescope, to I 0 I 1 I_0I_1 I 0 I 1 I − 1 I 2 = I 1 I 2 I_{-1}I_2=I_1I_2 I − 1 I 2 = I 1 I 2 ∑ j ≥ 1 j I j − 1 I j + 1 = κ 4 ( I 0 I 1 + I 1 I 2 ) \sum_{j\geq1}jI_{j-1}I_{j+1}=\frac{\kappa}{4}(I_0I_1+I_1I_2) ∑ j ≥ 1 j I j − 1 I j + 1 = 4 κ ( I 0 I 1 + I 1 I 2 )
∑ j ≥ 1 j 3 I j 2 = κ 2 4 ( I 0 2 + κ 2 I 1 ( I 0 − I 2 ) ) = κ 2 4 ( I 0 2 + I 1 2 ) . \sum_{j\geq1}j^3I_j^2=\frac{\kappa^2}{4}\left(I_0^2+\frac{\kappa}{2}I_1(I_0-I_2)\right)=\frac{\kappa^2}{4}(I_0^2+I_1^2). j ≥ 1 ∑ j 3 I j 2 = 4 κ 2 ( I 0 2 + 2 κ I 1 ( I 0 − I 2 ) ) = 4 κ 2 ( I 0 2 + I 1 2 ) . The coefficients of D \mathcal{D} D
Lemma 132. Write D ( κ ) = ∑ n ≥ 0 D n κ 2 n \mathcal{D}(\kappa)=\sum_{n\geq0}\mathcal{D}_n\kappa^{2n} D ( κ ) = ∑ n ≥ 0 D n κ 2 n D 0 = D 1 = 0 \mathcal{D}_0=\mathcal{D}_1=0 D 0 = D 1 = 0 m ≥ 1 m\geq1 m ≥ 1
D m + 1 = N m 4 m m ! ( m + 1 ) ! ( m + 1 ) ( m + 2 ) , \mathcal{D}_{m+1}=\frac{N_m}{4^m m!(m+1)!(m+1)(m+2)}, D m + 1 = 4 m m ! ( m + 1 )! ( m + 1 ) ( m + 2 ) N m , with N m = ( 2 m m ) ( 4 ( 2 m + 1 ) + ( m − 1 ) ( m + 1 ) ( m + 2 ) ) − 4 m ( m + 1 ) ( m + 2 ) N_m=\binom{2m}{m}\left(4(2m+1)+(m-1)(m+1)(m+2)\right)-4^m(m+1)(m+2) N m = ( m 2 m ) ( 4 ( 2 m + 1 ) + ( m − 1 ) ( m + 1 ) ( m + 2 ) ) − 4 m ( m + 1 ) ( m + 2 ) N m = 0 N_m=0 N m = 0 m ≤ 2 m\leq2 m ≤ 2 N m > 0 N_m>0 N m > 0 m ≥ 3 m\geq3 m ≥ 3
Proof. The coefficients. Multiplying the series of I 0 I_0 I 0 I 1 I_1 I 1
I 0 2 = ∑ m ≥ 0 α m κ 2 m , I 1 2 = ∑ m ≥ 0 β m κ 2 m + 2 , I_0^2=\sum_{m\geq0}\alpha_m\kappa^{2m},\qquad I_1^2=\sum_{m\geq0}\beta_m\kappa^{2m+2}, I 0 2 = m ≥ 0 ∑ α m κ 2 m , I 1 2 = m ≥ 0 ∑ β m κ 2 m + 2 , κ I 1 ( 2 κ ) = ∑ m ≥ 0 κ 2 m + 2 m ! ( m + 1 ) ! , \kappa I_1(2\kappa)=\sum_{m\geq0}\frac{\kappa^{2m+2}}{m!(m+1)!}, κ I 1 ( 2 κ ) = m ≥ 0 ∑ m ! ( m + 1 )! κ 2 m + 2 , where
α m = 1 4 m ∑ i = 0 m 1 i ! 2 ( m − i ) ! 2 , \alpha_m=\frac{1}{4^m}\sum_{i=0}^{m}\frac{1}{i!^2(m-i)!^2}, α m = 4 m 1 i = 0 ∑ m i ! 2 ( m − i ) ! 2 1 , β m = 1 4 m + 1 ∑ i = 0 m 1 i ! ( i + 1 ) ! ( m − i ) ! ( m − i + 1 ) ! . \beta_m=\frac{1}{4^{m+1}}\sum_{i=0}^{m}\frac{1}{i!(i+1)!(m-i)!(m-i+1)!}. β m = 4 m + 1 1 i = 0 ∑ m i ! ( i + 1 )! ( m − i )! ( m − i + 1 )! 1 . By Vandermonde’s identity, ∑ i ( m i ) 2 = ( 2 m m ) \sum_i\binom{m}{i}^2=\binom{2m}{m} ∑ i ( i m ) 2 = ( m 2 m ) ∑ i ( m i ) ( m + 2 i + 1 ) = ( 2 m + 2 m + 1 ) \sum_i\binom{m}{i}\binom{m+2}{i+1}=\binom{2m+2}{m+1} ∑ i ( i m ) ( i + 1 m + 2 ) = ( m + 1 2 m + 2 )
α m = ( 2 m m ) 4 m m ! 2 , β m = ( 2 m + 2 m + 1 ) 4 m + 1 m ! ( m + 2 ) ! . \alpha_m=\frac{\binom{2m}{m}}{4^m m!^2},\qquad\beta_m=\frac{\binom{2m+2}{m+1}}{4^{m+1}m!(m+2)!}. α m = 4 m m ! 2 ( m 2 m ) , β m = 4 m + 1 m ! ( m + 2 )! ( m + 1 2 m + 2 ) . The coefficient of κ 2 m + 2 \kappa^{2m+2} κ 2 m + 2 D \mathcal{D} D D m + 1 = 8 β m + 2 β m − 1 − α m − 1 / ( m ! ( m + 1 ) ! ) \mathcal{D}_{m+1}=8\beta_m+2\beta_{m-1}-\alpha_m-1/(m!(m+1)!) D m + 1 = 8 β m + 2 β m − 1 − α m − 1/ ( m ! ( m + 1 )!) β − 1 = 0 \beta_{-1}=0 β − 1 = 0 m = 0 m=0 m = 0 8 4 − 1 − 1 = 0 \frac{8}{4}-1-1=0 4 8 − 1 − 1 = 0 D 0 = 0 \mathcal{D}_0=0 D 0 = 0 D \mathcal{D} D O ( κ 2 ) O(\kappa^2) O ( κ 2 ) m ≥ 1 m\geq1 m ≥ 1 Δ = 4 m m ! ( m + 1 ) ! ( m + 1 ) ( m + 2 ) \Delta=4^m m!(m+1)!(m+1)(m+2) Δ = 4 m m ! ( m + 1 )! ( m + 1 ) ( m + 2 ) b = ( 2 m m ) b=\binom{2m}{m} b = ( m 2 m )
( 2 m + 2 m + 1 ) = 2 ( 2 m + 1 ) m + 1 b , \binom{2m+2}{m+1}=\frac{2(2m+1)}{m+1}b, ( m + 1 2 m + 2 ) = m + 1 2 ( 2 m + 1 ) b , 8 β m = 4 ( 2 m + 1 ) b Δ , α m = ( m + 1 ) ( m + 2 ) b Δ , 8\beta_m=\frac{4(2m+1)b}{\Delta},\qquad\alpha_m=\frac{(m+1)(m+2)b}{\Delta}, 8 β m = Δ 4 ( 2 m + 1 ) b , α m = Δ ( m + 1 ) ( m + 2 ) b , 2 β m − 1 = 2 m ( m + 1 ) ( m + 2 ) b Δ , 2\beta_{m-1}=\frac{2m(m+1)(m+2)b}{\Delta}, 2 β m − 1 = Δ 2 m ( m + 1 ) ( m + 2 ) b , 1 m ! ( m + 1 ) ! = 4 m ( m + 1 ) ( m + 2 ) Δ , \frac{1}{m!(m+1)!}=\frac{4^m(m+1)(m+2)}{\Delta}, m ! ( m + 1 )! 1 = Δ 4 m ( m + 1 ) ( m + 2 ) , and 2 m − ( m + 1 ) = m − 1 2m-(m+1)=m-1 2 m − ( m + 1 ) = m − 1 D m + 1 = N m / Δ \mathcal{D}_{m+1}=N_m/\Delta D m + 1 = N m /Δ
The sign. Directly, N 0 = N 1 = N 2 = 0 N_0=N_1=N_2=0 N 0 = N 1 = N 2 = 0 2 − 2 2-2 2 − 2 24 − 24 24-24 24 − 24 192 − 192 192-192 192 − 192 N 3 = 80 N_3=80 N 3 = 80 N 4 = 1140 N_4=1140 N 4 = 1140 N 5 = 10416 N_5=10416 N 5 = 10416 m ≥ 6 m\geq6 m ≥ 6 16 m ≤ 4 m ( 2 m m ) 2 16^m\leq4^m\binom{2m}{m}^2 1 6 m ≤ 4 m ( m 2 m ) 2 m ≥ 1 m\geq1 m ≥ 1 m = 1 m=1 m = 1 ( m + 1 ) ( 2 m + 2 m + 1 ) = 2 ( 2 m + 1 ) ( 2 m m ) (m+1)\binom{2m+2}{m+1}=2(2m+1)\binom{2m}{m} ( m + 1 ) ( m + 1 2 m + 2 ) = 2 ( 2 m + 1 ) ( m 2 m )
4 ( m + 1 ) ( 2 m + 2 m + 1 ) 2 = 16 ( 2 m + 1 ) 2 m + 1 ( 2 m m ) 2 ≥ 16 ( 2 m + 1 ) 2 4 m ( m + 1 ) 16 m ≥ 16 m + 1 . 4(m+1)\binom{2m+2}{m+1}^2=\frac{16(2m+1)^2}{m+1}\binom{2m}{m}^2\geq\frac{16(2m+1)^2}{4m(m+1)}16^m\geq16^{m+1}. 4 ( m + 1 ) ( m + 1 2 m + 2 ) 2 = m + 1 16 ( 2 m + 1 ) 2 ( m 2 m ) 2 ≥ 4 m ( m + 1 ) 16 ( 2 m + 1 ) 2 1 6 m ≥ 1 6 m + 1 . Since ( m − 1 ) 2 ≥ 4 m (m-1)^2\geq4m ( m − 1 ) 2 ≥ 4 m m ≥ 6 m\geq6 m ≥ 6 ( 2 m m ) 2 ( m − 1 ) 2 ≥ 16 m \binom{2m}{m}^2(m-1)^2\geq16^m ( m 2 m ) 2 ( m − 1 ) 2 ≥ 1 6 m ( 2 m m ) ( m − 1 ) ≥ 4 m \binom{2m}{m}(m-1)\geq4^m ( m 2 m ) ( m − 1 ) ≥ 4 m
N m = ( ( 2 m m ) ( m − 1 ) − 4 m ) ( m + 1 ) ( m + 2 ) + 4 ( 2 m + 1 ) ( 2 m m ) > 0. N_m=\left(\binom{2m}{m}(m-1)-4^m\right)(m+1)(m+2)+4(2m+1)\binom{2m}{m}>0. N m = ( ( m 2 m ) ( m − 1 ) − 4 m ) ( m + 1 ) ( m + 2 ) + 4 ( 2 m + 1 ) ( m 2 m ) > 0. Monotonicity of Bessel ratios
Lemma 133. For n ≥ 1 n\geq1 n ≥ 1 κ > 0 \kappa>0 κ > 0 I n + 2 ( κ ) / I n + 1 ( κ ) ≤ I 2 ( κ ) / I 1 ( κ ) I_{n+2}(\kappa)/I_{n+1}(\kappa)\leq I_2(\kappa)/I_1(\kappa) I n + 2 ( κ ) / I n + 1 ( κ ) ≤ I 2 ( κ ) / I 1 ( κ )
Proof. We use a lemma of Biernacki and Krzyż [19 ] : if f = ∑ a j x j f=\sum a_jx^j f = ∑ a j x j g = ∑ b j x j g=\sum b_jx^j g = ∑ b j x j a j / b j a_j/b_j a j / b j j j j f / g f/g f / g x > 0 x>0 x > 0
f ′ g − f g ′ = ∑ i , j ( i − j ) a i b j x i + j − 1 = 1 2 ∑ i , j ( i − j ) b i b j ( a i b i − a j b j ) x i + j − 1 ≤ 0. f'g-fg'=\sum_{i,j}(i-j)a_ib_jx^{i+j-1}
=\frac{1}{2}\sum_{i,j}(i-j)b_ib_j\left(\frac{a_i}{b_i}-\frac{a_j}{b_j}\right)x^{i+j-1}\leq0. f ′ g − f g ′ = i , j ∑ ( i − j ) a i b j x i + j − 1 = 2 1 i , j ∑ ( i − j ) b i b j ( b i a i − b j a j ) x i + j − 1 ≤ 0. Take x = κ 2 x=\kappa^2 x = κ 2 f = I n + 1 ( κ ) / κ n + 1 f=I_{n+1}(\kappa)/\kappa^{n+1} f = I n + 1 ( κ ) / κ n + 1 g = I 1 ( κ ) / κ g=I_1(\kappa)/\kappa g = I 1 ( κ ) / κ a j / b j = 2 − n ( j + 1 ) ! / ( j + n + 1 ) ! a_j/b_j=2^{-n}(j+1)!/(j+n+1)! a j / b j = 2 − n ( j + 1 )! / ( j + n + 1 )! j j j I n + 1 / ( κ n I 1 ) I_{n+1}/(\kappa^nI_1) I n + 1 / ( κ n I 1 ) κ \kappa κ
n + 1 κ + I n + 2 I n + 1 − n κ − 1 κ − I 2 I 1 = I n + 2 I n + 1 − I 2 I 1 ≤ 0. \frac{n+1}{\kappa}+\frac{I_{n+2}}{I_{n+1}}-\frac{n}{\kappa}-\frac{1}{\kappa}-\frac{I_2}{I_1}
=\frac{I_{n+2}}{I_{n+1}}-\frac{I_2}{I_1}\leq0. κ n + 1 + I n + 1 I n + 2 − κ n − κ 1 − I 1 I 2 = I n + 1 I n + 2 − I 1 I 2 ≤ 0. 
