Introduction
Wilson’s lattice gauge theory [69 ] 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 ∑ p Re Tr Q p ) \exp(N\beta\sum_p \operatorname{Re}\operatorname{Tr} Q_p) exp ( N β ∑ p Re Tr Q p ) Q p Q_p Q p p p p 1 ∣ β ∣ |\beta| ∣ β ∣ [56 , 60 ] .2 [40 ] begins where strong coupling ends and Bałaban’s weak-coupling analysis [4 , 5 , 6 ] takes over.
Two methods give explicit ranges. The Bakry–Émery method [2 , 3 ] of Shen, Zhu and Zhu [61 ] bounds the Hessian of the action; in [13 ] we showed that the sharp bound is 4 d = 16 4d=16 4 d = 16 ∣ β ∣ < 1 / 32 |\beta|<1/32 ∣ β ∣ < 1/32 [26 , 27 ] 3 4 [13 ] (Theorem 91 in Appendix § 1 ), and every link has eighteen partners, which gives β W < 2 / 9 \beta_W<2/9 β W < 2/9
Both limits come from the same count. A link of Z 4 \mathbb{Z}^4 Z 4 [24 ] .5
Theorem 1 (SU(2)). Let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) 0 ≤ β W < 1 3 0\leq\beta_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_W) − log ( 3 β W )
m ≥ − log ( 3 β W ) > 0. m\geq-\log(3\beta_W)>0. m ≥ − log ( 3 β W ) > 0. For − 1 3 < β W < 0 -\frac{1}{3}<\beta_W<0 − 3 1 < β W < 0
The step behind Theorem 1 is short once the line field is in place.7 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_W\geq\frac{1}{3} β W ≥ 3 1 β W = 1 3 \beta_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), so 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_W\leq0.345 0 ≤ β W ≤ 0.345 α ( β W ) \alpha(\beta_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_W) − 2 1 log α ( β W )
m ≥ − 1 2 log α ( β W ) > 0 ; m\geq-\frac{1}{2}\log\alpha(\beta_W)>0; m ≥ − 2 1 log α ( β W ) > 0 ; at β W = 1 3 \beta_W=\frac{1}{3} β W = 3 1 m > 0.0224 m>0.0224 m > 0.0224 − 0.345 < β W < 0 -0.345<\beta_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_W=\frac{1}{3} β W = 3 1 Theorem 1 has closed.
survey of Chatterjee [22 ] , his large-N N N N N N [23 ] and its planar extension by Basu and Ganguly [9 ] , the Langevin dynamics of Shen, Zhu and Zhu [61 ] , and the area law in an enlarged region by Cao, Nissim and Sheffield [21 ] . For finite gauge groups Adhikari and Cao proved correlation decay at weak coupling [1 ] . In two dimensions the continuum Yang–Mills measure has been constructed by Gross, King and Sengupta [34 ] , Driver [29 ] and Lévy [45 ] ; in four dimensions Bałaban’s renormalisation group, as expounded by Dimock [25 ] , gives ultraviolet stability in finite volume, and Magnen, Rivasseau and Senéor constructed the theory with an infrared cutoff [49 ] . None of these results gives a mass gap in physical units.
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 Q Q Q v = ( X e Q e ) v=(X_eQ_e) v = ( X e Q e ) X e = 0 X_e=0 X e = 0 e ∈ F e\in F e ∈ F
∣ Hess S 0 ( v , v ) ∣ ≤ 29 2 ∣ v ∣ 2 , \left|\operatorname{Hess}S_0(v,v)\right|\leq\frac{29}{2}|v|^2, ∣ Hess S 0 ( v , v ) ∣ ≤ 2 29 ∣ v ∣ 2 , against the sharp constant 16 16 16 G = SU ( N ) G=\operatorname{SU}(N) G = SU ( N ) ∣ β ∣ < 1 29 |\beta|<\frac{1}{29} ∣ β ∣ < 29 1 G = SO ( N ) G=\operatorname{SO}(N) G = SO ( N ) N ≥ 3 N\geq3 N ≥ 3 ∣ β ∣ < N − 2 58 N |\beta|<\frac{N-2}{58N} ∣ β ∣ < 58 N N − 2 Theorem 1 hold with the rate and gap of Theorem 5 . For SU ( 3 ) \operatorname{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 ) \operatorname{U}(N) U ( N ) ρ : G → U ( N ) \rho:G\to\operatorname{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 8 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\operatorname{U}(N) ρ ( G ) ⊂ U ( N )
Theorem 5 (Every compact simple group). Suppose that the Lie algebra of G G G κ G > 0 \kappa_G>0 κ G > 0
N ∣ β ∣ < 2 29 κ G . N|\beta|<\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|\beta|}-\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=\frac{h^\vee}{2\ell_\rho} κ G = 2 ℓ ρ h ∨ h ∨ h^\vee h ∨ ℓ ρ \ell_\rho ℓ ρ ρ \rho ρ
For SU ( N ) \operatorname{SU}(N) SU ( N ) κ G = N 2 \kappa_G=\frac{N}{2} κ G = 2 N SO ( N ) \operatorname{SO}(N) SO ( N ) κ G = N − 2 4 \kappa_G=\frac{N-2}{4} κ G = 4 N − 2 Theorem 5 contains the ranges of Theorem 4 and supplies their rate.9 ∣ β ∣ |\beta| ∣ β ∣ [56 Thm. 3.6] ; what is new is an explicit range for every group, past the range N ∣ β ∣ < κ G / 16 N|\beta|<\kappa_G/16 N ∣ β ∣ < κ G /16 16 16 16 β → 0 \beta\to0 β → 0 log κ G 6 N ∣ β ∣ \log\frac{\kappa_G}{6N|\beta|} log 6 N ∣ β ∣ κ G
Table 1. 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
No Bakry–Émery argument for the ungauged measure can pass ∣ β ∣ = 1 32 |\beta|=\frac{1}{32} ∣ β ∣ = 32 1 [13 Thm. C] ; the gauge fixing changes the measure, not the metric, which is why it can. For SU ( 2 ) \operatorname{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 SU ( 2 ) \operatorname{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 SU ( 3 ) \operatorname{SU}(3) SU ( 3 )
Theorem 6 (SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) Let G = SU ( 3 ) G=\operatorname{SU}(3) G = SU ( 3 ) 0 ≤ β W ≤ 3 4 0\leq\beta_W\leq\frac{3}{4} 0 ≤ β W ≤ 4 3 Theorem 1 hold, with decay at every rate below − log ( 1.33319 β W ) -\log(1.33319\beta_W) − log ( 1.33319 β W )
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 SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) b b b ν b \nu_b ν b b b b ν b \nu_b ν b
The plaquette blocks of Theorem 3 work for SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) Λ ( 2 β W ) \Lambda(2\beta_W) Λ ( 2 β W ) 2 Λ ( r ) < 1 2\Lambda(r)<1 2Λ ( r ) < 1 Theorem 6 they give a much larger gap, and they go past it.
Theorem 7 (SU ( 3 ) \operatorname{SU}(3) SU ( 3 ) Let G = SU ( 3 ) G=\operatorname{SU}(3) G = SU ( 3 ) 0 ≤ β W ≤ 19 25 = 0.76 0\leq\beta_W\leq\frac{19}{25}=0.76 0 ≤ β W ≤ 25 19 = 0.76 α 3 ( β W ) < 1 \alpha_3(\beta_W)<1 α 3 ( β W ) < 1 (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 α 3 ( 0.76 ) ≤ 0.99950 \alpha_3(0.76)\leq0.99950 α 3 ( 0.76 ) ≤ 0.99950 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 )
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 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.
gap
m ≥ − 1 2 log α 3 ( β W ) > 0 ; m \ge-\frac{1}{2}\log\alpha_3(\beta_W)>0; m ≥ − 2 1 log α 3 ( β W ) > 0 ; 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 Balaban’s regime, or a gap in physical units, which together are the Clay problem; § 2.3 says where each method stops and what remains of the four open problems of [13 ] . The obstruction proved in [14 ] , that a lattice gap uniform in lattice units forces trivial continuum two-point functions, applies to all of them.
The continuum step. § 2.4 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 ), also for observables smoothed by the spatial gradient flow (Proposition 46 ).
(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 47 ); that ratio is bounded above and below by ratios of correlations of one slice field at fixed physical separations (Proposition 50).
(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 55 ); for a smooth packet they give the ratio with an explicit rate and a reference vector of finite energy (Proposition 59).
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. The open inputs are measured as follows. 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 N N ( β W / log β W ) 1 / 4 (\beta_W/\log\beta_W)^{1/4} ( β W / log β W ) 1/4
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.
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 ).
Plan. Sec. II constructs the line field. Sec. III shows that a Dobrushin bound for the gauge-fixed specification gives the full conclusion of Theorem 1 . Secs. IV, V, VI, VII and VIII prove Theorems 1 –7 , and Sec. IX measures what is left; Sec. X proves (I1)–(I3) and measures the open inputs. 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 F), the facts on spectral measures and on the character expansion that the continuum step uses (Appendices G and § 2.4.8 ), 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.
II 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 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 v ∈ R v\in R v ∈ R π x ( s ) = s \pi_x(s)=s π x ( s ) = s π x ( v ) ≠ s \pi_x(v)\ne s π x ( v ) = s v ∈ R v\in R v ∈ 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 ν ] + [ π x ( ν ) = μ ] \nu] + [\pi_x(\nu) = \mu] 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
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 10
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 (a product of four finite intervals, with the links and plaquettes it contains), F ∩ E ( Λ ) F \cap E(\Lambda) F ∩ E ( Λ ) is a disjoint union of segments, hence a forest.
(iii) A link e = ( x , μ ) ∉ F e = (x,\mu) \notin F e = ( x , μ ) ∈ / F lies in six plaquettes. Four of them contain a link of F F F that shares an end point with e e e and these lie in the planes { μ , σ ( x ) } \{\mu,\sigma(x)\} { μ , σ ( x )} and { μ , π x ( μ ) } \{\mu,\pi_x(\mu)\} { μ , π x ( μ )} In the other two, which lie in the remaining plane { μ , ρ } \{\mu,\rho\} { μ , ρ } the link of F F F is opposite to e e e
(iv) Every link outside F F F shares a plaquette with exactly twelve other links outside F F F two in each of its plaquettes.
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 } 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\mathrm{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 . 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\left(N\beta\sum_{p\in P(\Lambda)}\operatorname{Re}\operatorname{Tr}Q_p^F\right) 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\left(N\beta\sum_{p\ni e}\operatorname{Re}\operatorname{Tr}Q_p^F\right) 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 ) \mathrm{SU}(2) SU ( 2 ) S 3 S^3 S 3 D = 2 D=2 D = 2 S U ( N ) \mathrm{SU}(N) SU ( 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 ′ ) |\operatorname{Cov}_\omega(O,O')|\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
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 τ \tau τ 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 3 e − λ R C_T R^3e^{-\lambda R} C T R 3 e − λ R R R R T T T Λ \Lambda Λ O O O 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 [56 ] , 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 a \tau_a τ a O O O ∣ ω ( ϑ ( O ) τ a O ) − ω ( O ) ∣ 2 ∣ ≤ C O e − λ t |\omega(\vartheta(O)\tau_aO)-\omega(O)|^2|\le C_Oe^{-\lambda t} ∣ ω ( ϑ ( O ) τ a O ) − ω ( O ) ∣ 2 ∣ ≤ C O e − λ t O O O L 2 ( ω ) L^2(\omega) L 2 ( ω ) G E G^E G E L 2 L^2 L 2 [ 0 , e − λ ] [0,e^{-\lambda}] [ 0 , e − λ ] λ < − log α \lambda<-\log\alpha λ < − log α
Figure 4. (a) The line field in the slice x 3 = 0 x_3=0 x 3 = 0 F F F 0 , 1 , 2 0,1,2 0 , 1 , 2 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
The proposition replaces the torus limits and the uniqueness of the ungauged Gibbs measure used in [13 Thm. F] by the free-boundary limit, which has all the properties the reconstruction of the transfer matrix needs.11
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_e s_p)=\frac{1}{2}\operatorname{Tr}Q_p Re ( Q e s p ) = 2 1 Tr Q p F F F N β Re Tr Q p = 4 β ⟨ Q e , s ˉ p ⟩ N\beta\operatorname{Re}\operatorname{Tr}Q_p=4\beta\langle Q_e,\bar{s}_p\rangle N β Re Tr Q p = 4 β ⟨ Q e , s ˉ 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 ˉ p b=4\beta\sum_{p\ni e}\bar{s}_p b = 4 β ∑ p ∋ e s ˉ p e ′ e' e ′ δ \delta δ b b b 4 ∣ β ∣ δ 4|\beta|\delta 4∣ β ∣ δ Theorem 91 ), 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 ∣ 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 □ \square □
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 ∣ 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 ∣ □ \square □
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 [13 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\geq0}\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
A block plaquette is a plaquette of the set B \mathcal{B} B tbl:4 of Appendix C, and its block is the set of its three active links.
Lemma 12. The set B \mathcal{B} B 2 Z 4 2\mathbb{Z}^{4} 2 Z 4
the blocks partition the active links;
every plaquette p ∉ B p\notin\mathcal{B} p ∈ / B
every active link lies in exactly five plaquettes outside B \mathcal{B} B B \mathcal{B} B
if an active link j j j B B B B \mathcal{B} B B B B j j j
Proof. All four statements are invariant under 2 Z 4 2\mathbb{Z}^{4} 2 Z 4 Z 4 \mathbb{Z}^{4} Z 4 ( Z / P Z ) 4 (\mathbb{Z}/P\mathbb{Z})^{4} ( Z / P Z ) 4 P ≥ 6 P\geq6 P ≥ 6
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_{1}y_{2}y_{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) \mu_{b}(dy)\propto\exp\left(\sum_{i=1}^{3}\langle y_{i},b_{i}\rangle+4\beta\operatorname{Re}(y_{1}y_{2}y_{3})\right)\,dy_{1}\,dy_{2}\,dy_{3}, \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) 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^{3}g(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^{2}M),\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-Lipschitz f f f h ∈ R 4 h\in\mathbb{R}^{4} h ∈ R 4
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 ∣
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^{2}M)|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 u=x u = x 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 ⟩ ) = 1 4 d d t ν c + t h ( f ) ∣ t = 0 ≤ 1 4 ∣ h ∣ \operatorname{Cov}_{\nu}(f,\langle x,h\rangle)=\frac{1}{4}\frac{d}{dt}\nu_{c+th}(f)\big|_{t=0}\leq\frac{1}{4}|h| Cov ν ( f , ⟨ x , h ⟩) = 4 1 d t d ν c + t h ( f ) t = 0 ≤ 4 1 ∣ h ∣ sd ν ( f ) ≤ τ ∣ h ∣ \operatorname{sd}_{\nu}(f)\leq\sqrt{\tau}|h| sd ν ( f ) ≤ τ ∣ h ∣ 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 ˉ ) = ∇ g ( c 0 ) + 4 β Hess g ( c 0 ) u + N ( u ) \nabla g(c_{0}+4\beta\bar{u})=\nabla g(c_{0})+4\beta\operatorname{Hess}g(c_{0})u+N(u) ∇ g ( c 0 + 4 β u ˉ ) = ∇ 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
so ∣ N ( u ) − N ( u ′ ) ∣ ≤ 16 β 2 M ∣ u − u ′ ∣ |N(u)-N(u')|\le16\beta^{2}M|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^{2}M|k| 16 β 2 M ∣ k ∣ f f f 16 β 2 M ∣ k ∣ 16\beta^{2}M|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'|\le20|\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'})\le 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 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 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 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 ∣ [7 ] 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 [41 ] 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 Y B \mathcal{Y}^{B} Y B B B B Y B B \mathcal{Y}^{B}_{B} Y B B B B B Y F \mathcal{Y}^{F} Y F
C B j = sup W 1 ( Y B B ( ⋅ ∣ Q ) , Y B B ( ⋅ ∣ Q ′ ) ) ∣ Q j − Q j ′ ∣ C_{Bj}=\sup\frac{W_1(\mathcal{Y}^{B}_{B}(\cdot\mid Q),\mathcal{Y}^{B}_{B}(\cdot\mid Q'))}{|Q_j-Q_j'|} C B j = sup ∣ Q j − Q j ′ ∣ W 1 ( Y B B ( ⋅ ∣ Q ) , Y 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 80 of Appendix B in place of Lemma 78, 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 ∣ ∣ 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 i b_i b i i i i 4 ∣ β ∣ η 4|\beta|\eta 4∣ β ∣ η ∑ B C B j ≤ 40 ∣ β ∣ K \sum_B C_{Bj}\le40|\beta|K ∑ B C B j ≤ 40∣ β ∣ K β W = 0.345 \beta_W=0.345 β W = 0.345 w ≤ 0.0054559 w\le0.0054559 w ≤ 0.0054559 s ≤ 0.252746 s\le0.252746 s ≤ 0.252746 C ≤ 0.050243 C\le0.050243 C ≤ 0.050243 K ≤ 0.2890609 K\le0.2890609 K ≤ 0.2890609 α ≤ 0.99729 \alpha\le0.99729 α ≤ 0.99729 β W = 1 3 \beta_W=\frac{1}{3} 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 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\mathrm{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 ) \mathrm{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},\qquad Y_2=Q_{e_1}X_{e_2}Q_{e_1}^{*},\qquad Y_3=Q_{e_4}X_{e_3}Q_{e_4}^{*},\qquad Y_4=X_{e_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 [13 Lemma 1] ). In a box two perpendicular ends at y y y
The term + 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 [13 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\mathrm{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\leq\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
∑ 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)\leq\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\geq0 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\leq\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 [13 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 ) \mathrm{U}(N) U ( N ) G ⊂ U ( N ) G\subset\mathrm{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 ) \mathrm{U}(N) U ( N )
Lemma 21. Let G ⊂ U ( N ) G\subset\mathrm{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}_{\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=-\frac{1}{4}[[X,Y],Z] R ( X , Y ) Z = − 4 1 [[ X , Y ] , Z ] [52 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
=\frac{1}{4}\sum_i\lvert[X,E_i]\rvert^2=-\frac{1}{4}\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 ⟨ ⋅ , ⋅ ⟩ B=-4\kappa\langle\cdot,\cdot\rangle B = − 4 κ ⟨ ⋅ , ⋅ ⟩ κ \kappa κ Ric = κ ⟨ ⋅ , ⋅ ⟩ \operatorname{Ric}=\kappa\langle\cdot,\cdot\rangle Ric = κ ⟨ ⋅ , ⋅ ⟩ 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 [39 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)+(\overline{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 ) [62 ] . 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_{ee'}(a,b)=H^p_{e'e}(b,a) H e e ′ p ( a , b ) = H e ′ e p ( b , a ) ∣ H e e ′ p ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ \lvert H^p_{ee'}(a,b)\rvert\leq\lvert a\rvert\lvert b\rvert ∣ 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 ∣ \lvert\operatorname{Tr}(abW)\rvert\leq\lVert a\rVert_{\mathrm{HS}}\lVert bW\rVert_{\mathrm{HS}}=\lvert a\rvert\lvert b\rvert ∣ 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\lvert\beta\rvert<\frac{2}{29}\kappa_G N ∣ β ∣ < 29 2 κ G μ F A \mu_F^A μ F A K = κ G − 29 2 N ∣ β ∣ > 0 K=\kappa_G-\frac{29}{2}N\lvert\beta\rvert>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 [3 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\leq c_p\leq1 0 ≤ c p ≤ 1 ∣ Hess S ( v , v ) ∣ ≤ 29 2 ∣ v ∣ 2 \lvert\operatorname{Hess}S(v,v)\rvert\leq\frac{29}{2}\lvert v\rvert^2 ∣ Hess S ( v , v )∣ ≤ 2 29 ∣ v ∣ 2 v v v G A ′ G^{A'} G A ′ ∣ Hess S e e ′ ( a , b ) ∣ ≤ ∣ a ∣ ∣ b ∣ \lvert\operatorname{Hess}S_{ee'}(a,b)\rvert\leq\lvert a\rvert\lvert b\rvert ∣ Hess S e e ′ ( a , b )∣ ≤ ∣ a ∣ ∣ b ∣ S A = ∑ p ∈ P ( A ) Re Tr Q p F S_A=\sum_{p\in P(A)}\operatorname{Re}\operatorname{Tr}Q_p^F S A = ∑ p ∈ P ( A ) 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\lvert\nabla_e f\rvert^2 Γ w ( f ) = ∑ e w e ∣ ∇ e f ∣ 2 Γ \Gamma Γ w = 1 w=1 w = 1
Lemma 24. Let λ ≥ 0 \lambda\geq0 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 \lvert d_e-d_{e'}\rvert\leq1 ∣ 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\lvert\beta\rvert\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 ) \mathop{\mathrm{\Gamma}}\nolimits^w(P_tf)\leq 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\geq0 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 ‾ ∇ g + Ric ( ∇ g , ∇ g ) \nabla\Delta g=\overline{\Delta}\nabla g+\operatorname{Ric}(\nabla g,\nabla g) ∇Δ g = Δ ∇ g + Ric ( ∇ g , ∇ 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(\lvert\nabla\nabla_e g\rvert^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\lvert\beta\rvert\lvert v\rvert^2 − 2 29 N ∣ β ∣ ∣ v ∣ 2 cosh λ 2 − 1 \cosh\frac{\lambda}{2}-1 cosh 2 λ − 1 N ∣ β ∣ N\lvert\beta\rvert N ∣ β ∣ 2 ∣ v e ∣ ∣ v e ′ ∣ ≤ ∣ v e ∣ 2 + ∣ v e ′ ∣ 2 2\lvert v_e\rvert\lvert v_{e'}\rvert\leq\lvert v_e\rvert^2+\lvert v_{e'}\rvert^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\lvert\beta\rvert\lvert v\rvert^2 12 ( cosh 2 λ − 1 ) N ∣ β ∣ ∣ v ∣ 2 Ric G ≥ κ G \operatorname{Ric}_G\geq\kappa_G Ric G ≥ κ G ∣ v ∣ 2 = Γ w ( g ) \lvert v\rvert^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)\geq 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 ( P t − s Γ w ( f ) ) \varphi(s)=P_s^{}(P_{t-s}\Gamma^w(f)) φ ( s ) = P s ( P t − s Γ w ( f )) 0 ≤ s ≤ t 0\leq s\leq 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))\geq2K_\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)\leq 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 ∥ ∞ . \left|\operatorname{Cov}_{\mu}(f,g)\right| \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 ∥ ∇ f ∥ ∞ 2 = sup ∑ e ∣ ∇ e f ∣ 2 \|\nabla f\|_\infty^2=\sup\sum_e|\nabla_e f|^2 ∥∇ f ∥ ∞ 2 = sup ∑ e ∣ ∇ 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 P t w ( f ) ≤ e − λ D e − 2 K λ t ∥ ∇ f ∥ ∞ 2 |\nabla_eP_t f|^2\le w_e^{-1}P_t^w(f)\le e^{-\lambda D}e^{-2K_\lambda t}\|\nabla f\|_\infty^2 ∣ ∇ e P t f ∣ 2 ≤ w e − 1 P t w ( f ) ≤ e − λ D e − 2 K λ t ∥∇ f ∥ ∞ 2 ∣ Γ ( P t f , g ) ∣ ≤ e − λ D / 2 e − K λ t ∥ ∇ f ∥ ∞ ∑ e ∈ B g ∥ ∇ e g ∥ ∞ |\Gamma(P_t f,g)|\le e^{-\lambda D/2}e^{-K_\lambda t}\|\nabla f\|_\infty\sum_{e\in B_g}\|\nabla_e g\|_\infty ∣Γ ( P t f , g ) ∣ ≤ e − λ D /2 e − 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 ( f , g ) ∣ ≤ C f , g e − λ D / 2 , \left|\operatorname{Cov}_{\mu_\Lambda^F}(f,g)\right|\le C_{f,g}e^{-\lambda D/2}, Cov μ Λ F ( 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 = μ Λ F \mu_0=\mu_\Lambda^F μ 0 = μ Λ F Λ \Lambda Λ μ 1 = μ Λ ′ F \mu_1=\mu_{\Lambda'}^F μ 1 = μ Λ ′ F 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 P(\Lambda')\setminus 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 ) 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 τ Δ O e^{\tau\Delta}O e τ Δ O G E G^E G E ω ( O ) \omega(O) ω ( O ) W Λ ( O ) W_\Lambda(O) W Λ ( O )
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 S U ( 3 ) \mathrm{SU}(3) SU ( 3 ) X , Y ∈ s u ( 3 ) X,Y\in\mathfrak{su}(3) X , Y ∈ su ( 3 ) K = const ( 3 ) K=\operatorname{const}(3) K = const ( 3 ) A ( M ) = 1 2 ( M − M ∗ ) A(M)=\frac{1}{2}(M-M^*) A ( M ) = 2 1 ( M − M ∗ ) P P P Ric = 3 2 \operatorname{Ric}=\frac{3}{2} Ric = 2 3 λ F = 8 3 \lambda_F=\frac{8}{3} 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}(U b)}\,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) Properties. Invariance and, for β ≥ 0 \beta\ge0 β ≥ 0 O O O e τ Δ O e^{\tau\Delta}O e τ Δ O Ric G ≥ 0 \operatorname{Ric}_G\ge0 Ric G ≥ 0 [3 Sect. 3.3] , so the bound of the first step passes to ω \omega ω
Proof of the rest of Theorem 4 . By Lemma 21 and Table I, k S U ( N ) = N 2 k_{\mathrm{SU}(N)}=\frac{N}{2} k SU ( N ) = 2 N k S O ( N ) = N − 2 4 k_{\mathrm{SO}(N)}=\frac{N-2}{4} k 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 \mathbb{R}^4\cong\mathbb{C}^2\otimes\mathbb{C}^2 R 4 ≅ C 2 ⊗ C 2 1 2 \frac{1}{2} 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\ldots 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 = 13.97401 h_F=13.97401 h F = 13.97401 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
Since K K K 0 0 0 I , ω I , ω 2 I I,\omega I,\omega^2 I I , ω I , 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 ) = Re Tr ( U X 2 b ) \operatorname{Hess}V(X,X)=\operatorname{Re}\operatorname{Tr}(UX^2b) Hess V ( X , X ) = Re Tr ( 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 \varphi_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 SU ( 3 ) \operatorname{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 : SU ( 3 ) → s u ( 3 ) f:\operatorname{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 ad = 1 2 ( C π + C σ ) − 3 2 . \nabla^*\nabla=\frac{1}{2}(C_\pi+C_\sigma)-\frac{1}{4}C_{\operatorname{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 ) = 3 2 ( p 2 + q 2 + p q + 3 p + 3 q ) C(p,q)=\frac{3}{2}(p^2+q^2+pq+3p+3q) C ( p , q ) = 2 3 ( p 2 + q 2 + pq + 3 p + 3 q ) ( p , q ) (p,q) ( p , q ) [30 ] . 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 19 6 \frac{19}{6} 6 19 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=\varphi_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 ∫ ∣ ω ∣ 2 = ∫ ∣ ∇ ω ∣ 2 + ∫ ( ( Ric − Hess V ) ( ω , ω ) ) \lambda\int|\omega|^2=\int|\nabla\omega|^2+\int((\operatorname{Ric}-\operatorname{Hess}V)(\omega,\omega)) λ ∫ ∣ ω ∣ 2 = ∫ ∣∇ ω ∣ 2 + ∫ (( 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 \mathcal A=\nabla\eta A = ∇ η B = 1 2 d V ⊗ η \mathcal 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\mathcal A,\mathcal B\rangle=\frac{1}{4}\int\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 ⟩ \|\mathcal A-\mathcal B\|^2=\|\frac{5}{8}\mathcal A-\mathcal B\|^2+\frac{39}{64}\|\mathcal A\|^2-\frac{3}{4}\langle\mathcal A,\mathcal 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 \|\mathcal A-\mathcal 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 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 ∈ SU ( 3 ) W\in\operatorname{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_kc_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 ) ∑ d k = 0 \sum d_k=0 ∑ 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 e 1 μ + e 2 = 0 3\mu^2-2e_1\mu+e_2=0 3 μ 2 − 2 e 1 μ + e 2 = 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 4 \frac{1}{4} 4 1 V V V 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 13
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 SU ( 3 ) \operatorname{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}(\varphi_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 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\phi_h u = s ϕ h ϕ h + L u = ( 1 − s λ F ) ϕ h + s Γ \phi_h+Lu=(1-s\lambda_F)\phi_h+s\Gamma ϕ h + Lu = ( 1 − s λ F ) ϕ h + s Γ Γ = ⟨ ∇ V , ∇ ϕ h ⟩ \Gamma=\langle\nabla V,\nabla\phi_h\rangle Γ = ⟨ ∇ V , ∇ ϕ h ⟩ s = 1 / λ F s=1/\lambda_F s = 1/ λ F ϕ h \phi_h ϕ h β W = 45 64 \beta_W=\frac{45}{64} β W = 64 45 § 2.3 ). We take u u u
Let P 2 \mathcal P_2 P 2 u i j u_{ij} u ij U U U u ‾ i j \overline{u}_{ij} u ij U ‾ \overline U U ( B k ) (B_k) ( B k ) 1 2 ( m + m ‾ ) \frac{1}{2}(m+\overline m) 2 1 ( m + m ) 1 2 i ( m − m ‾ ) \frac{1}{2i}(m-\overline m) 2 i 1 ( m − m ) { m , m ‾ } \{m,\overline m\} { m , m } m ≠ m ‾ m\ne\overline m m = m m m m m = m ‾ m=\overline m m = m 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 = 1 18 ∑ m C k m h m B k ∈ P 2 , q_h=\sum_{k=1}^{18}\sum_m C_{km}h_mB_k\in\mathcal P_2, q h = k = 1 ∑ 18 m ∑ 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_kb_k\phi_{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=\phi_h+Lq_h=\phi_h+\Delta q_h+\sum_kb_k\langle\nabla\phi_{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|\nabla q_h|^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_aw_aX_afX_ag,\qquad\Delta=\sum_aw_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\phi_h=-\frac{8}{3}\phi_h Δ ϕ h = − 3 8 ϕ h
Orbits of staple sums
Lemma 31. Let V , W ∈ SU ( 3 ) V,W\in\operatorname{SU}(3) V , W ∈ SU ( 3 )
(i) 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 ‾ ) = L h ‾ ( b ) L_h(\overline b)=L_{\overline h}(b) L h ( b ) = L h ( b )
(ii) 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|z_3| a 1 ≥ a 2 ≥ ∣ z 3 ∣ Im z 3 ≥ 0 \operatorname{Im}z_3\ge0 Im z 3 ≥ 0
(iii) A diagonal matrix lies in K K K T T T SU ( 3 ) \operatorname{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 ) \phi_h(U)=\phi_{V^{-1}hW^{-1}}(WUV) ϕ h ( U ) = ϕ V − 1 h W − 1 ( W U V ) b ‾ \overline b b U ↦ U ‾ U\mapsto\overline U U ↦ U 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 ) SU ( 3 ) \operatorname{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 SU ( 3 ) Re Tr ( c t ) = max T Re Tr ( c t ) \max_{\operatorname{SU}(3)}\operatorname{Re}\operatorname{Tr}(ct)=\max_T\operatorname{Re}\operatorname{Tr}(ct) max SU ( 3 ) Re Tr ( c t ) = max T Re Tr ( c t ) c c c c c c conv T \operatorname{conv}T conv T K K K
Since every b ∈ r K b\in rK b ∈ r K r r r
Λ ( r ) = sup ∥ h ∥ = 1 { sup b ∈ B r L h ( b ) } , \Lambda(r)=\sup_{\|h\|=1}\left\{\sup_{b\in\mathcal B_r}L_h(b)\right\}, Λ ( r ) = ∥ h ∥ = 1 sup { b ∈ B r sup L h ( b ) } , where B r \mathcal B_r B 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|x_3+iy_3| 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 SU ( 3 ) \operatorname{SU}(3) SU ( 3 )
Z ( D , D ′ ) = E H exp ( 1 2 Tr ( U D ) + 1 2 Tr ( U ‾ D ′ ) ) . Z(D,D')=\mathbb E_H\exp\left(\frac{1}{2}\operatorname{Tr}(UD)+\frac{1}{2}\operatorname{Tr}(\overline U D')\right). Z ( D , D ′ ) = E H exp ( 2 1 Tr ( U D ) + 2 1 Tr ( U 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^{-|\lambda|}/\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 ‾ \overline U U E H [ u α u ‾ β e Tr ( U b ) ] = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ‾ ) \mathbb E_H[u^\alpha\overline u^\beta e^{\operatorname{Tr}(U b)}]=2^{|\alpha|+|\beta|}\partial_D^\alpha\partial_{D'}^\beta Z(b,\overline b) E H [ u α u β e Tr ( U b ) ] = 2 ∣ α ∣ + ∣ β ∣ ∂ D α ∂ D ′ β Z ( b , b ) u i j u_{ij} u ij D j i D_{ji} D j i
Proof. We use ( Tr X ) n = ∑ λ ⊢ n f λ s λ ( X ) (\operatorname{Tr}X)^n=\sum_{\lambda\vdash n}f^\lambda s_\lambda(X) ( Tr X ) n = ∑ λ ⊢ n f λ s λ ( X ) f λ f^\lambda f λ λ \lambda λ [63 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)^k s_\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 ‾ = ( U − 1 ) T \overline{U}=(U^{-1})^{\mathsf{T}} U = ( U − 1 ) T χ p , q ( U ‾ D ′ ) = χ p , q ( U − 1 D ′ T ) \chi_{p,q}(\overline{U}D')=\chi_{p,q}(U^{-1}D'^{\mathsf{T}}) χ p , q ( U D ′ ) = χ p , q ( U − 1 D ′ T ) [20 Ch. 2] ,
E H [ χ ρ ( U A ) χ ρ ′ ( U − 1 B ) ] = δ ρ ρ ′ χ ρ ( A B ) d ρ , \mathbb{E}_{H}\left[\chi_{\rho}(UA)\chi_{\rho'}(U^{-1}B)\right]=\delta_{\rho\rho'}\frac{\chi_{\rho}(AB)}{d_{\rho}}, E H [ χ ρ ( U A ) χ ρ ′ ( U − 1 B ) ] = δ ρ ρ ′ d ρ χ ρ ( 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}(Ub)=\frac{1}{2}\operatorname{Tr}(Ub)+\frac{1}{2}\operatorname{Tr}(\overline{U}\overline{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\operatorname{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\operatorname{dim}\lambda=3^n ∑ λ ⊢ n f λ dim λ = 3 n ∑ n > 34 ( 2 3 ∥ D ∥ ) n / n ! \sum_{n>34}\left(\frac{2}{3}\|D\|\right)^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 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 ) 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 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 ] [38 ] 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 ) 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 ) F-I_1I_2I_3I_4F=\sum_k I_1\cdots I_{k-1}(F-I_kF) F − I 1 I 2 I 3 I 4 F = ∑ k I 1 ⋯ I k − 1 ( F − I k F ) k k k F F F ℓ 2 \ell^2 ℓ 2 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^2F=((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 ) F-I_kF\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 ) γ ( r ) + 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{\frac{e^{\varsigma/8}\max_i\lambda_{\mathcal{A}}(b_i)}{\gamma(r)}}+
\sqrt{\frac{e^{\varsigma/8}}{\gamma(r)}\left(\max_i\lambda_{\mathcal{V}}(b_i)+\frac{e_N}{\min_i Z(b_i)}\right)}. \tag*{(13)} h sup L h ( b ) ≤ γ ( r ) e ς /8 max i 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_{\mathcal{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 I I I [36 , 41 ] .
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\ge0 θ ≥ 0 ν ∣ ∇ q ∣ 2 + θ Var ( X ) / γ \nu|\nabla q|^2+\theta\operatorname{Var}(X)/\gamma ν ∣∇ 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 > max I r ℓ \min_{Q}\ell>\max_{\mathcal{I}_r}\ell min Q ℓ > max I r ℓ conv T \operatorname{conv}T 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 2 r B Q < 1 2rB_Q<1 2 r B Q < 1 B Q B_Q B Q γ ( r ) = 187 128 \gamma(r)=\frac{187}{128} γ ( r ) = 128 187 r / 16 r/16 r /16 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 5 shows the cover, and Appendix E the error analysis.
Figure 5. 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 ∣ |x_3+\mathrm{i}y_3| ∣ 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 2 r B Q < 1.03 2rB_Q<1.03 2 r B Q < 1.03 r / 8 r/8 r /8 r / 16 r/16 r /16 R r \mathcal{R}_r R r
Proof of Theorem 6 . Proposition 34 gives
Λ ( 3 2 ) ≤ 0.999887 3 < 0.333296. \Lambda\left(\frac{3}{2}\right)\leq\frac{0.999887}{3}<0.333296. Λ ( 2 3 ) ≤ 3 0.999887 < 0.333296. For β W ≤ 3 4 \beta_W\leq\frac{3}{4} β W ≤ 4 3 2 β W ≤ 3 2 2\beta_W\leq\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\leq4\beta_W\Lambda\left(\frac{3}{2}\right)<1.333183\beta_W\leq\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\geq0 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 C 3 × 3 \mathbb{C}^{3\times3} C 3 × 3 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 , (14) \mu_b(dy)\propto\exp\left(\sum_i\operatorname{Re}\operatorname{Tr}(y_i b_i)+N\beta\operatorname{Re}\operatorname{Tr}(y_1y_2y_3)\right)dy, \tag*{(14)} μ b ( d y ) ∝ exp ( i ∑ Re Tr ( y i b i ) + N β Re Tr ( y 1 y 2 y 3 ) ) d y , (14) 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}}\leq5N\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
A 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 , (15) K_3=\frac{(1+N\beta\Lambda(r))s}{1-C},\qquad\alpha_3=10N\beta K_3, \tag*{(15)} K 3 = 1 − C ( 1 + N β Λ ( r )) s , α 3 = 10 N β K 3 , (15) 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 \kappa(dx)\propto\exp(\operatorname{Re}\operatorname{Tr}(xa)+g(c_0+N\beta x))dx κ ( d x ) ∝ exp ( Re Tr ( x a ) + g ( c 0 + N β x )) d x on S U ( 3 ) \mathrm{SU}(3) SU ( 3 )
For every 1-Lipschitz 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)\leq 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\leq 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\leq 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)\leq\Lambda(r)\lVert h\rVert Cov ν ( f , ϕ h ) ≤ Λ ( r ) ∥ h ∥ f f f d d d sd ν f ≤ γ − 1 / 2 \operatorname{sd}_{\nu}f\leq\gamma^{-1/2} sd ν f ≤ γ − 1/2 sd ν ϕ h ≤ τ ∥ h ∥ \operatorname{sd}_{\nu}\phi_h\leq\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 ) + Hess g ( c 0 ) u + N ( u ) \nabla g(c_0+N\beta u)=\nabla g(c_0)+\operatorname{Hess}g(c_0)u+N(u) ∇ g ( c 0 + N β u ) = ∇ g ( c 0 ) + Hess g ( c 0 ) u + N ( u ) ϕ h \phi_h ϕ h ∥ h ∥ ≤ N β τ ∥ k ∥ \lVert h\rVert\leq 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\leq\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 ∥ \sqrt{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 [37 ] , and the covariance of two Lipschitz functions under κ \kappa κ e w / γ e^w/\gamma e w / γ
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'})\leq 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 3 ′ y_3' y 3 ′ y 3 − 1 b 2 y_3^{-1}b_2 y 3 − 1 b 2 ∥ 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\leq5N\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\leq s\varepsilon+Cd_3 d 1 ≤ s ε + C d 3 d 3 ≤ C d 1 d_3\leq 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\leq(1+N\beta\Lambda(r))(d_1+d_3)\leq K_3\varepsilon ∣ μ b ( f ) − μ b ′ ( f )∣ ≤ ( 1 + N β Λ ( r )) ( d 1 + d 3 ) ≤ K 3 ε
B The constants and the proof of Theorem 7
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\leq0.182126 ∣ 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 r \mathcal{R}_r R r g g g φ \varphi 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 § 2 G. On each box Q Q Q φ \varphi 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 \varphi 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}(\varphi_h)\leq\nu_b[(\varphi_h-\langle k,h\rangle)^2]\leq e^{\varepsilon/8}\max_i\nu_{b_i}[(\varphi_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\geq0 f ≥ 0 Z ( b ) ≤ ∑ i ω i Z ( b i ) Z(b)\leq\sum_i\omega_iZ(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)\geq 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=(\varphi_h-\nu_b\varphi_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}(\varphi_h)\geq e^{-\varepsilon/8}\min_i\operatorname{Var}_{\nu_{b_i}}(\varphi_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 \|\delta\|\leq\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}(\varphi_v)-2\delta_v\operatorname{Cov}(\varphi_u,\varphi_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[(\varphi-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 |\nu_b(G)|\leq\frac{\max_i|\nu_{b_i}(G)|+\varepsilon\max_i\nu_{b_i}|G|}{1-\varepsilon} ∣ ν b ( G ) ∣ ≤ 1 − ε max i ∣ ν b i ( G ) ∣ + ε max i ν b i ∣ G ∣ with 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=(\varphi_u-k_u)(\varphi_v-k_v)^2 G = ( φ u − k u ) ( φ v − k v ) 2 ∣ ν b i ( G ) ∣ ≤ ∥ T b i ∥ |\nu_{b_i}(G)|\leq\|T_{b_i}\| ∣ 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}|G|\leq2\sqrt{3}\,\nu_{b_i}[(\varphi_v-k_v)^2] ν b i ∣ G ∣ ≤ 2 3 ν b i [( φ v − k v ) 2 ] ∥ U − k ∥ H S ≤ 2 3 \|U-k\|_{\mathrm{HS}}\leq2\sqrt{3} ∥ U − k ∥ HS ≤ 2 3 § 2 G, and the maxima over these boxes are the numbers stated (Appendix C).
Proof of Theorem 7 . Let first β W ≤ 3 4 \beta_W\leq\frac{3}{4} β W ≤ 4 3 r = 2 β W ≤ 3 2 r=2\beta_W\leq\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})\leq\frac{0.999887}{3} Λ ( 2 3 ) ≤ 3 0.999887 γ = γ ( 3 2 ) = 187 128 \gamma=\gamma(\frac{3}{2})=\frac{187}{128} γ = γ ( 2 3 ) = 128 187 (15) is increasing in β W \beta_W β W 3 4 < β W ≤ 19 25 \frac{3}{4}<\beta_W\leq\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})\leq\frac{1.029967}{3.04} Λ ( 25 38 ) ≤ 3.04 1.029967 γ ( 38 25 ) = 4643 4320 \gamma(\frac{38}{25})=\frac{4643}{4320} γ ( 25 38 ) = 4320 4643 j j j η j \eta_j η j η \eta η b i b_i b i N β η N\beta\eta N β η ∑ B C B j ≤ 10 N β K 3 = α 3 ( β W ) \sum_B C_Bj\leq10N\beta K_3=\alpha_3(\beta_W) ∑ B C B j ≤ 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\leq0.017935 w ≤ 0.017935 s ≤ 0.34088 s\leq0.34088 s ≤ 0.34088 C ≤ 0.043520 C\leq0.043520 C ≤ 0.043520 K 3 ≤ 0.386090 K_3\leq0.386090 K 3 ≤ 0.386090 α 3 ≤ 0.96521 \alpha_3\leq0.96521 α 3 ≤ 0.96521 β W = 0.72 \beta_W=0.72 β W = 0.72 α 3 ≤ 0.91815 \alpha_3\leq0.91815 α 3 ≤ 0.91815 β W = 0.76 \beta_W=0.76 β W = 0.76 w ≤ 0.018609 w\leq0.018609 w ≤ 0.018609 s ≤ 0.34673 s\leq0.34673 s ≤ 0.34673 C ≤ 0.045770 C\leq0.045770 C ≤ 0.045770 K 3 ≤ 0.39454 K_3\leq0.39454 K 3 ≤ 0.39454 α 3 ≤ 0.99950 \alpha_3\leq0.99950 α 3 ≤ 0.99950
Remark 39. Theorem 6 ends where 2 r Λ ( r ) 2r\Lambda(r) 2 r Λ ( r ) 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 0.0177 0.0177 0.0177 β W ≥ 0.715 \beta_W\geq0.715 β W ≥ 0.715 12 10 \frac{12}{10} 10 12 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 β W ≈ 0.78 \beta_W \approx0.78 β W ≈ 0.78
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 [28 ] , pass it: plaquette blocks reach β W = 0.345 \beta_W = 0.345 β W = 0.345
Curvature. Remark 26 locates the end of the curvature method in this gauge between ∣ β ∣ = 1 29 |\beta| = \frac{1}{29} ∣ β ∣ = 29 1 1 126.25 \frac{1}{126.25} 126.25 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_W \approx0.77 β W ≈ 0.77 u = s φ h u = s\varphi_h u = s φ h B r \mathcal{B}_r B 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_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 ( F i g .6 ) r = 1.54 (Fig. 6) r = 1.54 ( F i g .6 ) r = 1.5 r = 1.5 r = 1.5 β W = 3 4 \beta_W = \frac{3}{4} β W = 4 3 Λ ( 2 β W ) \Lambda(2\beta_W) Λ ( 2 β W )
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 2 29 \frac{2}{29} 29 2 G G G N ∣ β ∣ = 2 29 κ G N|\beta| = \frac{2}{29}\kappa_G N ∣ β ∣ = 29 2 κ G
The four open problems. Our earlier paper [13 ] listed four problems between the strong-coupling results and the Clay problem [40 ] : (1) uniqueness and a gap at intermediate fixed coupling; (2) the crossover from strong coupling to Balaban’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_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_W \approx2.2 β W ≈ 2.2 β W ≈ 6 \beta_W \approx6 β W ≈ 6 [14 ] 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 [4 , 16 ] .14
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 ) (16) \mathbb{E}F(\chi) \le e^{C(\|\chi\|_1+\|\chi\|_2^2)} \tag*{(16)} E F ( χ ) ≤ e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) (16) for all real χ ∈ S ( R d ) \chi\in\mathcal{S}(\mathbb{R}^d) χ ∈ S ( R d ) p = 2 p = 2 p = 2 [33 Sect. 6.1, (6.1.5)] . It is this hypothesis under which positivity across coordinate and diagonal hyperplanes is shown in [65 ] 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 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) \ge c|x-y|^{-2\Delta}(\log\frac{1}{|x-y|})^{-k} S 2 ( x , y ) ≥ c ∣ x − y ∣ − 2Δ ( log ∣ x − y ∣ 1 ) − k 0 < ∣ x − y ∣ ≤ r 0 0 < |x-y| \le r_0 0 < ∣ x − y ∣ ≤ r 0 c > 0 c > 0 c > 0 k ≥ 0 k \ge0 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 \ge1 + \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 \le2(e^{C(\|\chi\|_1+\|\chi\|_2^2)}-1) E F ( χ ) 2 ≤ 2 ( e C ( ∥ χ ∥ 1 + ∥ χ ∥ 2 2 ) − 1 ) φ , ψ ∈ C ∞ ( R d ) \varphi,\psi\in C^\infty(\mathbb{R}^d) φ , ψ ∈ C ∞ ( R d ) r 0 2 \frac{r_0}{2} 2 r 0 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 \|\varphi_\varepsilon\|_1 \le\varepsilon^{d/2}\|\varphi\|_1 ∥ φ ε ∥ 1 ≤ ε d /2 ∥ φ ∥ 1 ε ≤ 1 \varepsilon\le1 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 ≤ ∣ e u − e v ∣ ≤ ε R \varepsilon\le|eu-ev| \le\varepsilon R ε ≤ ∣ e u − e v ∣ ≤ εR ε R ≤ r 0 \varepsilon R \le 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 R ) − 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 \\
&\ge cR^{-2\Delta}\varepsilon^{d-2\Delta}\left(\log\frac{1}{\varepsilon R}\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 εR 1 ) − k ∥ φ ∥ 1 ∥ ψ ∥ 1 , which tends to infinity as $\varepsilon \to 0. \quad \square
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 15 [66 ] states for the smeared action density, has 2 Δ = 8 > 4 2\Delta=8>4 2Δ = 8 > 4 k = 2 k=2 k = 2 (16) , and rotation invariance for it has to come from another source, as in [66 ] , 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 (16) suits fields of dimension at most d / 2 d/2 d /2
Figure 6. 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
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}{|p|^2+m^2}\,dm. C ( p ) = ∫ 0 1 ∣ p ∣ 2 + m 2 m 2 d m . Then F F F (16) 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^2dm m 2 d m m m m [33 Sect. 6.2] ; a positive combination of reflection positive covariances is reflection positive, and so is the Gaussian measure with such a covariance [33 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_0^1m^2\,dm\int\frac{|\widehat{g}(k)|^2}{2\omega}e^{-\omega s}\left(\int_0^\infty h(t)e^{-\omega t}\,dt\right)^2dk, ⟨ 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{|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_0s}\lVert v\rVert^2 e − m 0 s ∥ v ∥ 2 0 < η < m 0 0<\eta<m_0 0 < η < m 0 m , ∣ k ∣ ≤ η / 2 m,|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^{[t/a]}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 [14 ] 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 Balaban’s small-field analysis [4 , 6 ] , which the conditional construction of [66 ] 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 [66 Sect. 9.4] reduces a gap to a threshold condition on one box, uniformly in the cutoff, as stated there, the condi-
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_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
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 [14 ] 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 [28 ] , 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 are named at the end: the cube estimate and the existence of the limit.
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 ≠ ∅ ReTr Q p F ) \exp\left(N\beta\sum_{p\cap V\ne\varnothing}\operatorname{ReTr}Q_p^F\right) exp ( N β ∑ p ∩ V = ∅ ReTr 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 7 shows the cubes.
Figure 7. (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_{\mathrm{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 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 ) □ \square □
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 T_k T k K k \mathcal{K}_k K k Ω k \Omega_k Ω k 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\left(O_k(f)\right), ⟨ 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 nontrivial if H ≠ C Ω \mathcal{H}\ne\mathbb{C}\Omega H = C Ω
The Osterwalder–Schrader reconstruction [54 , 55 ] 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 \mathbb{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 ⟨ Ψ F , T k t Ψ H ⟩ = ω k ( ϑ ( F ‾ ) τ t H ) \langle\Psi_F,\mathbb{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 ) ⟨ 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 \mathbb{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,\mathbb{T}_k^t\varphi_k\rangle
=\omega_k\bigl(\vartheta(\overline{O})\tau_tO\bigr)-|\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 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_*\tau}\mu_\varphi([0,\infty)) ∫ e − τ E d μ φ ( E ) ≤ e − m ∗ τ μ φ ([ 0 , ∞ )) for all τ ≥ 0 ; i f \tau\ge0; if τ ≥ 0 ; i f η > 0 \eta>0 η > 0 μ φ ( [ 0 , m ∗ − η ] ) e − ( m ∗ − η ) τ \mu_\varphi([0,m_*-\eta])e^{-(m_*-\eta)\tau} μ φ ([ 0 , m ∗ − η ]) e − ( m ∗ − η ) τ τ \tau τ μ φ \mu_\varphi μ φ [ m ∗ , ∞ ) [m_*,\infty) [ m ∗ , ∞ ) φ ∈ Ω ⊥ \varphi\in\Omega^\perp φ ∈ Ω ⊥ Ω ⊥ ≠ { 0 } \Omega^\perp\ne\{0\} Ω ⊥ = { 0 } [ m ∗ , ∞ ) [m_*,\infty) [ m ∗ , ∞ )
The theorem bypasses the obstruction of [14 ] : 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 [14 ] the hypothesis implies the gap part (R2) of Hypothesis R, with m = m ∗ m=m_* m = m ∗
Smoothed observables
The observables O k ( f ) O_k(f) O k ( f ) [47 ] .16 [48 ] . Flow in all four directions mixes the two sides of the reflection plane, but the spatial flow of [59 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{\mathrm{d}}{\mathrm{d}s}U_e=-\nabla_e\sum_{p\ni e,\ p\ \mathrm{spatial}}(N-\operatorname{Re}\operatorname{Tr}U_p) 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 46. (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) \le 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 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 ) ≤ ∣ s ∣ ( κ 0 d e ′ + c 1 ∑ e ′ ′ ∼ e ′ d e ′ ′ ) d_{e'}(s)\le|s|\left(\kappa_0d_{e'}+c_1\sum_{e''\sim e'}d_{e''}\right) d e ′ ( s ) ≤ ∣ 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'}\le 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!\le e^x x^{R+1}/(R+1)! ∑ k > R x k / k ! ≤ e x x R + 1 / ( R + 1 )! c = 12 c 1 c=12c_1 c = 12 c 1 κ = κ 0 + c 1 \kappa=\kappa_0+c_1 κ = κ 0 + c 1
(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 t Ψ H ⟩ = ω ( ϑ ( F ˉ ) τ t ( H ) ) \langle\Psi_F,T^t\Psi_H\rangle=\omega(\vartheta(\bar F)\tau_t(H)) ⟨ Ψ F , T t Ψ H ⟩ = ω ( ϑ ( F ˉ ) τ t ( H )) 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 ˉ ) τ t H ) = ω ( ϑ ( F ∘ Φ s ‾ ) τ t ( H ∘ Φ s ) ) (\Phi_s)_*\omega(\vartheta(\bar F)\tau_tH)=\omega(\vartheta(\overline{F\circ\Phi_s})\tau_t(H\circ\Phi_s)) ( Φ s ) ∗ ω ( ϑ ( F ˉ ) 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 ) Z k ∑ x a k 4 f ( a k x ) ( O τ ( x ) − ω ( O τ ( x ) ) ) Z_k\sum_xa_k^4f(a_kx)(\mathcal{O}_\tau(x)-\omega(\mathcal{O}_\tau(x))) Z k ∑ x a k 4 f ( a k x ) ( O τ ( x ) − ω ( O τ ( x ))) O τ ( x ) \mathcal{O}_\tau(x) O τ ( x ) [48 ] , and Theorem 45 applies to them. Theorem 47 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 0 ∈ C c ( R 4 ) f_0\in C_c(\mathbb{R}^4) f 0 ∈ 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 0 ( 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_0(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 0 ( a k x ) ( Φ x ∘ Φ s k − ω k ( Φ x ∘ Φ s k ) ) . C k ( σ ) = ω k ( ϑ ( A ˉ k ) τ σ / a k A k ) C_k(\sigma)=\omega_k\left(\vartheta(\bar A_k)\tau_{\sigma/a_k}A_k\right) C k ( σ ) = ω k ( ϑ ( A ˉ k ) τ σ / a k A k ) for dyadic σ ≥ 0 \sigma\ge0 σ ≥ 0 σ / a k \sigma/a_k σ / a k k k k C k ( 0 ) ≥ 0 C_k(0)\ge0 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 S \mathcal{S} S 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 ∈ S y\in\mathcal{S} y ∈ S 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 47. (i) For y , y ′ ∈ S y,y'\in\mathcal{S} y , y ′ ∈ S 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(\vartheta(\overline{A_k^y})A_k^{y'}) K ( y , y ′ ) = lim Z k 2 ω k ( ϑ ( A k y ) A k y ′ ) y , y ′ ∈ S y,y'\in\mathcal{S} y , y ′ ∈ S 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 ) σ > 0 \sigma>0 σ > 0 μ \mu μ [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ]
ρ ( σ ) = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) ( σ > 0 ) , \rho(\sigma)=\int_{[0,\infty]}e^{-\sigma E}\,\mathrm{d}\mu(E)\qquad(\sigma>0), ρ ( σ ) = ∫ [ 0 , ∞ ] e − σ E d μ ( E ) ( σ > 0 ) , the integrand being 0 0 0 E = ∞ E=\infty E = ∞
Figure 8. The packet f 0 f_0 f 0 Theorem 47 , 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 )
(iii) Along such a subsequence , let T \mathcal{T} T f 0 ( ⋅ − y ) f_0(\cdot-y) f 0 ( ⋅ − y ) y ∈ D y\in\mathcal{D} y ∈ D f ∗ ∈ C c ( R 4 ) f_*\in C_c(\mathbb{R}^4) f ∗ ∈ C c ( R 4 ) { t > 0 } \{t>0\} { t > 0 } O k O_k O k O k ( f 0 ( ⋅ − y ) ) = Z k A k y O_k(f_0(\cdot-y))=Z_kA_k^y O k ( f 0 ( ⋅ − y )) = Z k A k y O k ( f ∗ ) = 1 O_k(f_*)=1 O k ( f ∗ ) = 1 ( ω k , O k ) (\omega_k,O_k) ( ω k , O k ) 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 σ > 0 \sigma>0 σ > 0 ρ ( σ ) > 0 \rho(\sigma)>0 ρ ( σ ) > 0 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 σ > 0 \sigma>0 σ > 0
(v) If the hypothesis of Theorem 45 holds, then μ \mu μ [ m ∗ , ∞ ] [m_*,\infty] [ m ∗ , ∞ ] ρ ( σ ) ≤ e − m ∗ σ \rho(\sigma)\leq e^{-m_*\sigma} ρ ( σ ) ≤ e − m ∗ σ
Proof. By Proposition 46(ii) each Φ x ∘ Φ σ k \Phi_x\circ\Phi_{\sigma_k} Φ x ∘ Φ σ k A k y A_k^y A k y ≥ 1 \geq1 ≥ 1 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 |\langle Z_k\Psi_{A_k^y},Z_k\Psi_{A_k^{y'}}\rangle|\leq1 ∣ ⟨ 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\leq T_k\leq1 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_jc_jf_0(\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_yc_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>0 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 ) 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 ) v ) ≤ B ( v , v ) B(T(\sigma)v,T(\sigma)v)\leq B(v,v) B ( T ( σ ) v , T ( σ ) v ) ≤ B ( v , v ) B ( v , T ( σ ) v ) ≥ 0 B(v,T(\sigma)v)\geq0 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 ( 0 , y ′ ) ) (T(t/2)\delta_{(0,y')}) ( T ( t /2 ) δ ( 0 , 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\geq0 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(\cdot-y))=\delta_y ψ ( f 0 ( ⋅ − y )) = δ y ψ ( f ∗ ) = Ω \psi(f_*)=\Omega ψ ( f ∗ ) = Ω ψ \psi ψ 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 ( A k y ‾ ) = ω k ( τ σ / a k A k y ) = 0 \omega_k(\overline{A_k^y})=\omega_k(\tau_{\sigma/a_k}A_k^y)=0 ω k ( 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
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 Theorem 55 reduces them to tree bounds); (E) requires all of these.
The denominator 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 47 (iv) is about the smeared packets A k A_k A k
When the ratio is bounded below, renormalising at the separation σ \sigma σ 0 0 0 μ k \mu_k μ k Theorem 47 .
Proposition 48. (i) Suppose 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 > 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))\leq c^{-1}e^{-\sigma\Lambda},\qquad\int E^m\,d\mu'_k\leq 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 ′ μ \mu μ [ 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\leq 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)\geq e^{-2C\sigma} C k ( σ ) / C k ( 0 ) ≥ e − 2 C σ C a k ≤ 1 2 Ca_k\leq\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))\leq C(a_k+\Lambda^{-1}) μ k ([ Λ , ∞ )) ≤ C ( a k + Λ − 1 ) Λ > 0 \Lambda>0 Λ > 0
Proof. (i) We have Ψ A k ′ = ∏ k σ / 2 a k Ψ A k \Psi_{A'_k}=\prod_k^{\sigma/2a_k}\Psi_{A_k} Ψ A k ′ = ∏ 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)\geq c C k ( σ ) / C k ( 0 ) ≥ c e − σ E e^{-\sigma E} e − σ E E = ∞ E=\infty E = ∞ E ≥ 0 E\geq0 E ≥ 0 e − σ E ≤ e − σ Λ e^{-\sigma E}\leq e^{-\sigma\Lambda} e − σ E ≤ e − σ Λ E m e − σ E ≤ ( m / e σ ) m E^m e^{-\sigma E}\leq(m/e\sigma)^m E m e − σ E ≤ ( m / e σ ) m E ≥ 0 E\geq0 E ≥ 0 e − t E e^{-tE} e − tE μ k ′ \mu'_k μ k ′ ∣ e − t E − e − s E ∣ ≤ ∣ t − s ∣ E |e^{-tE}-e^{-sE}|\leq|t-s|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\geq\left(\int\lambda\,d\nu_k\right)^n\geq(1-Ca_k)^n\geq 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)\geq-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\geq\int(1-e^{-a_kE})\,d\mu_k\geq(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 , ∞ )) 1 − e − x ≥ x / ( 1 + x ) 1-e^{-x}\geq 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 49, with g g g a k a_k a k T k = A T_k=A T k = A Ψ A k = ∑ n w n T k n u \Psi_{A_k}=\sum_n w_nT_k^nu Ψ A k = ∑ n w n T 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 g ( a k x ) u=a_k g(a_kx) u = a k g ( a k x ) ( 1 − T k ) Ψ A k = ∑ n ( w n − w n − 1 ) T k n u (1-T_k)\Psi_{A_k}=\sum_n(w_n-w_{n-1})T_k^nu ( 1 − T k ) Ψ A k = ∑ n ( w n − w n − 1 ) T k n u ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ a k ∥ g ′ ∥ L 1 ∥ u ∥ / ∥ Ψ A k ∥ \langle\Psi_k,(1-T_k)\Psi_k\rangle\leq a_k\|g'\|_{L^1}\|u\|/\|\Psi_{A_k}\| ⟨ Ψ k , ( 1 − T k ) Ψ k ⟩ ≤ a k ∥ g ′ ∥ L 1 ∥ u ∥/∥ Ψ A k ∥ u u u
In a periodic spatial box the assumption C k ( 0 ) > 0 C_k(0)>0 C k ( 0 ) > 0 G = S U ( N ) G=SU(N) G = S U ( 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 47 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 49. 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 [46 ] ; 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 [46 , 56 ] . 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 φ 0 > 0 \varphi_0>0 φ 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_xh(a_kx)\Phi_x\circ\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 n p ( A ) F c ψ 0 \Psi_{A_k}=a_k^np(A)F^c\psi_0 Ψ A k = a k n 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_xh(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 Theorem 47 in a box; it says nothing about the ratio of part (iv), on which nontriviality depends.
The ratio of Theorem Theorem 47 (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)\left(\Phi_{(0,x)}\circ\Phi_{s_k}-\omega_k(\Phi_{(0,x)}\circ\Phi_{s_k})\right) 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 r r / a k u k ⟩ = ω k ( ϑ ( U ˉ k ) τ r / a k ( U k ) ) D_k(r)=\langle u_k,T_r^{r/a_k}u_k\rangle=\omega_k\left(\vartheta(\bar U_k)\tau_{r/a_k}(U_k)\right) D k ( r ) = ⟨ u k , T r r / a k u k ⟩ = ω k ( ϑ ( U ˉ k ) τ 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 50. 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<\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 ( 0 ) C_k(\sigma')\geq c^{\sigma'/\sigma}C_k(0) C k ( σ ′ ) ≥ c σ ′ / σ C k ( 0 ) 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 c ∈ ( 0 , 1 ] c\in(0,1] 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 D k ( 2 t 1 ) D_k(2t_0+\sigma')\geq cD_k(2t_1) D k ( 2 t 0 + σ ′ ) ≥ c 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 0 0 0 n ≥ 0 n\geq0 n ≥ 0 Ψ τ n F = T k n Ψ F \Psi_{\tau_nF}=\mathsf{T}_k^n\Psi_F Ψ τ n F = T k n Ψ F ω k ( ϑ ( G ˉ ) τ n F ) \omega_k(\vartheta(\bar G)\tau_nF) ω k ( ϑ ( G ˉ ) τ n F ) Ψ G \Psi_G Ψ G ω k \omega_k ω k A k = ∑ n w n τ n U k A_k=\sum_nw_n\tau_nU_k A k = ∑ n w n τ n U k 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 w n = 0 w_n=0 w n = 0 n 0 ≤ n ≤ n 1 n_0\leq n\leq n_1 n 0 ≤ n ≤ n 1 Ψ A k = ∑ n w n T k n u k \Psi_{A_k}=\sum_nw_n\mathsf{T}_k^n u_k Ψ A k = ∑ n w n T k n u k ν \nu ν T k \mathsf{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_nw_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\geq1 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\leq\lambda\leq1 0 ≤ λ ≤ 1 ∥ w ∥ 1 λ n 1 ≤ W ( λ ) ≤ ∥ w ∥ 1 λ n 0 \|w\|_1\lambda^{n_1}\leq W(\lambda)\leq\|w\|_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\leq1 λ ≤ 1 ρ = ν \rho=\nu ρ = ν ρ = W 2 ν \rho=W^2\nu ρ = W 2 ν β ≤ r ′ \beta\leq r' β ≤ r ′ D k ( β ) ≥ D k ( r ′ ) ≥ c D k ( r ) ≥ c D k ( α ) D_k(\beta)\geq D_k(r')\geq cD_k(r)\geq 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 ) (\log D_k(\beta)-\log D_k(\alpha))/(\beta-\alpha)\geq(\log D_k(r')-\log D_k(r))/(r'-r)\geq\log c/(r'-r) ( log D k ( β ) − log D k ( α )) / ( β − α ) ≥ ( log D k ( r ′ ) − log D k ( r )) / ( r ′ − r ) ≥ log c / ( r ′ − r ) 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 2t_0 2 t 0 2 t 1 2t_1 2 t 1 a k a_k a k 2 a k n 0 ≥ 2 t 0 2a_kn_0\geq2t_0 2 a k n 0 ≥ 2 t 0 2 a k n 1 ≤ 2 t 1 2a_kn_1\leq2t_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)}
\leq\frac{C_k(\sigma)}{C_k(0)}
\leq\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\geq 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 ′ / σ C k ( 0 ) C_k(\sigma')\geq c^{\sigma'/\sigma}C_k(0) C k ( σ ′ ) ≥ c σ ′ / 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 47 (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'w i t h with w i t h
and by (iii) a uniform bound for one packet gives one for every packet 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 ) = ( 1 − 4 − k ) 2 − k r + 4 − k 2 − r D_k(r)=(1-4^{-k})2^{-kr}+4^{-k}2^{-r} D k ( r ) = ( 1 − 4 − k ) 2 − k r + 4 − k 2 − r D k ( 4 ) / D k ( 3 ) → 1 2 D_k(4)/D_k(3)\to\frac{1}{2} 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 51. 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 ∞ ( R 3 ) h\in C^\infty(\mathbb{R}^3) h ∈ 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 \widehat p^2=\sum_i(2a_k^{-1}\sin(a_kp_i/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 \cosh(a_k\omega)=1+a_k^2\widehat p^2/2 cosh ( a k ω ) = 1 + a k 2 p 2 /2 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\le6 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=\frac{1}{2} r = 2 1 r ′ = 1 r'=1 r ′ = 1
Remark 52. The inequality also holds along the right tuning in two-dimensional SU ( N ) \operatorname{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 ReTr U p ) \prod_p\exp\left(\frac{\beta_k}{N}\operatorname{ReTr}U_p\right) ∏ p exp ( N β k ReTr 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 ReTr U e^{\frac{\beta}{N}\operatorname{ReTr}U} e N β ReTr 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'\cdots V_L'=W_1(V_1W_2V_1^{-1})\cdots 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 , \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 − 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 86 ) 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 N ( χ F + χ F ‾ ) \frac{\beta}{N}(\chi_F+\overline{\chi_F}) N β ( χ F + χ F ) 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[17 ] at the unique maximum U = 1 U=1 U = 1 ReTr U \operatorname{ReTr}U ReTr U U = e X U=e^X U = e X ReTr e X = N + 1 2 Tr X 2 + O ( ∣ X ∣ 4 ) \operatorname{ReTr}e^X=N+\frac{1}{2}\operatorname{Tr}X^2+O(|X|^4) ReTr e X = N + 2 1 Tr X 2 + O ( ∣ X ∣ 4 ) ReTr X 3 = 0 \operatorname{ReTr}X^3=0 ReTr X 3 = 0 Tr ρ λ ( X ) 2 = − 2 C 2 ( λ ) \operatorname{Tr}\rho_\lambda(X)^2=-2C_2(\lambda) Tr ρ λ ( X ) 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-NC_2(\lambda)/\beta+O(\beta^{-2}) c λ ( β ) = 1 − N C 2 ( λ ) / β + O ( β − 2 ) C 2 ( N 2 − 1 ) = 2 N C_2(N^2-1)=2N C 2 ( N 2 − 1 ) = 2 N Lemma 87 ). 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\|f\|^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}{\|f\|^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 \widehat 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 ( N 2 − 1 ) / ( 4 N ) e^{-(r'-r)E_{F,k}}\to e^{-(r'-r)g^2\ell(N^2-1)/(4N)} e − ( r ′ − r ) E F , k → e − ( r ′ − r ) g 2 ℓ ( N 2 − 1 ) / ( 4 N ) [29 , 34 , 45 ] , 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 expectation 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 53. 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 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 . Figure 9. Two-dimensional SU(N) (Remark [52 ] ). (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 10. 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
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 4 ∑ w ∈ { x B , y B } ( d f ( w ) + ∑ ν = 1 d f ( w − e ν ) ) , \alpha_B=\frac{1}{4}\sum_{w\in\{x_B,y_B\}}\left(d f(w)+\sum_{\nu=1}^{d}f(w-e_\nu)\right), α B = 4 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})\geq2\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\leq2d^2\sum_x f(x)^2 Var F ≤ 2 d 2 ∑ x f ( x ) 2
(iii) Let g ≥ 0 g\geq0 g ≥ 0 R d \mathbb{R}^d R d ∫ g 2 > 0 \int g^2>0 ∫ g 2 > 0 n ≥ 1 n\geq1 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}\geq2^{1-d},\qquad\limsup_{n\to\infty}\frac{\operatorname{Var}F_n}{n^d\int g^2}\leq2d^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)\geq(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})\geq\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(-\frac{1}{4}(2d+\mu^2)u_B^2+c_Bu_B) ∏ 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 + c o n s t \alpha_Bu_B^2+\beta_Bu_B=\alpha_Bv_B^2+(2\alpha_Bm_B+\beta_B)v_B+\mathrm{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 ( α v 2 + γ v ) = 2 α 2 σ 4 + γ 2 σ 2 \operatorname{Var}(\alpha v^2+\gamma v)=2\alpha^2\sigma^4+\gamma^2\sigma^2 Var ( α v 2 + γ v ) = 2 α 2 σ 4 + γ 2 σ 2 Var ( F ∣ H ) ≥ 2 σ 4 ∑ B α B 2 \operatorname{Var}(F\mid\mathcal{H})\geq2\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 \\
&\leq\sum_{x,y}\bigl(f(x)^2+f(y)^2\bigr)\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 \widehat 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)}{\widehat 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(\widehat p^2+\mu^2)^{-2}\leq1 ∑ 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 \widehat p^2 p 2 x x x
(iii) If Lip g = K \operatorname{Lip}g=K Lip g = 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-d\,g(2z/n)|\leq3dK/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\geq\sum_z(d\,g(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 4 = 4 / ( 2 d + μ 2 ) 2 → d − 2 \sigma^4=4/(2d+\mu^2)^2\to d^{-2} σ 4 = 4/ ( 2 d + μ 2 ) 2 → d − 2 ∑ 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 ))
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 ‾ k ) A k ) C_k(0)=\omega_k(\vartheta(\overline{A}_k)A_k) C k ( 0 ) = ω k ( ϑ ( A k ) A k )
only between the support of ϑ f 0 \vartheta f_0 ϑ f 0 f 0 f_0 f 0 Figure 11 shows the blocks between the reflection plane and the packet.
Figure 11. 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
Higher correlations
Theorem 47 controls two renormalised fields at a time, and part (i) of it gives nothing for more: Z k Z_k Z k Φ ∘ Φ s k \Phi_\circ\Phi_{s_k} Φ ∘ Φ s k Φ \Phi Φ ϕ k ( x ) = Φ ∘ Φ s k \phi_k(x)=\Phi_\circ\Phi_{s_k} ϕ k ( x ) = Φ ∘ Φ s k ω k T \omega_k^T ω k T ω k \omega_k ω k f ∈ C c ( R 4 ) f\in C_c(\mathbb{R}^4) f ∈ 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)\bigl(\phi_k(x)-\omega_k(\phi_k(x))\bigr), 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 = A k ( f ˉ ) \bar A_k=A_k(\bar f) A ˉ k = 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\geq2 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^T\bigl(\phi_k(x_1);\ldots;\phi_k(x_n)\bigr)\right|\leq 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 4 W k C_k(0)\geq c\,r_k^2a_k^4W_k C k ( 0 ) ≥ c r k 2 a k 4 W k c > 0 c>0 c > 0 k k k
(T3) Physical volume. B ∗ = sup k a k 4 W k < ∞ B_*=\sup_k a_k^4W_k<\infty B ∗ = sup k a k 4 W k < ∞
The amplitude r k r_k r k [56 ] shows why it is needed. For a compact 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 54. Under (T1), for n ≥ 2 n\geq2 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^T\bigl(A_k(f_1);\ldots;A_k(f_n)\bigr)\right|
\leq 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\geq2 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 1 1 1 W k W_k W k x 1 x_1 x 1 ∣ Q ∣ k a − 4 |Q|k_a^{-4} ∣ Q ∣ k a − 4 n n − 2 n^{n-2} n n − 2
Theorem 55. 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^T\left(Z_kA_k(f_1),\ldots,Z_kA_k(f_n)\right)\right|
\le K_n n^{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_j Z_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}^{n}Z_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 47 along the subsequence is K ( y , y ′ ) = S 2 ( f 0 ( ⋅ − y ) ∘ θ , f 0 ( ⋅ − y ′ ) ) K(y,y')=S_2(f_0(\cdot-y)\circ\theta,f_0(\cdot-y')) K ( y , y ′ ) = S 2 ( f 0 ( ⋅ − y ) ∘ θ , f 0 ( ⋅ − y ′ )) 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 α ‾ c β S ( ∏ j f α , j ‾ ∘ θ ⋅ ∏ j f β , j ) ≥ 0 , \sum_{\alpha,\beta}\overline{c_\alpha}c_\beta S\left(\prod_j\overline{f_{\alpha,j}}\circ\theta\cdot\prod_j f_{\beta,j}\right)\ge0, α , β ∑ c 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(\theta\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\ge0 t ≥ 0 H \mathcal{H} H e − t H e^{-tH} e − t H H ≥ 0 H\ge0 H ≥ 0 H H H σ > 0 \sigma>0 σ > 0 C k ( σ ) ≥ 1 2 C k ( 0 ) C_k(\sigma)\ge\frac12C_k(0) C k ( σ ) ≥ 2 1 C k ( 0 ) k k k Theorem 47 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\ne0 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 n ! n! n ! 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}\le B_*^{n/2-1} ( a k 4 W k ) n /2 − 1 ≤ B ∗ n /2 − 1 n ≥ 2 n\ge2 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\le(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 ( ∏ Z k A k ( f j ) ) \omega_k(\prod Z_kA_k(f_j)) ω k ( ∏ 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_j Z_kA_k(f_j)\right)-\omega_k\left(\prod_j Z_kA_k(g_j)\right)\right|
\le M_n(Q)\sum_i\lVert f_i-g_i\rVert_\infty\prod_{j\ne i}\max(\lVert f_j\rVert_\infty,\lVert g_j\rVert_\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 46 (i)) give ω k ( ∏ Z k A k ( f j ( ⋅ − y ) ) ) = ω k ( ∏ Z k A k ( f j ) ) \omega_k(\prod Z_kA_k(f_j(\mathord\cdot-y)))=\omega_k(\prod Z_kA_k(f_j)) ω k ( ∏ Z k A k ( f j ( ⋅ − y ))) = ω k ( ∏ 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(\mathord\cdot-y_m) f j ( ⋅ − y m ) f j ( ⋅ − y ) f_j(\mathord\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(\mathord\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 ) ‾ ) = A k ( f ‾ ∘ θ k ) \vartheta(\overline{A_k(f)})=A_k(\overline{f}\circ\theta_k) ϑ ( A k ( f ) ) = A k ( f ∘ θ k ) P k = ∑ β c β ∏ j Z k A k ( f β , j ) P_k=\sum_\beta c_\beta\prod_j Z_kA_k(f_{\beta,j}) P k = ∑ β c β ∏ j Z k A k ( f β , j ) R k = ∑ α c α ‾ ∏ j Z k A k ( f α , j ∘ θ ) R_k=\sum_\alpha\overline{c_\alpha}\prod_j Z_kA_k(f_{\alpha,j}\circ\theta) R k = ∑ α c α ∏ j Z k A k ( f α , j ∘ θ ) ϑ ( P k ‾ ) \vartheta(\overline{P_k}) ϑ ( P k ) R k R_k R k θ \theta θ θ k \theta_k θ k ω k ( ϑ ( P k ‾ ) P k ) ≥ 0 \omega_k(\vartheta(\overline{P_k})P_k)\ge0 ω k ( ϑ ( P k ) P k ) ≥ 0 ∣ ω k ( ϑ ( P k ‾ ) P k ) − ω k ( R k P k ) ∣ \left|\omega_k(\vartheta(\overline{P_k})P_k)-\omega_k(R_kP_k)\right| ω k ( ϑ ( P 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}\lVert f_{\alpha,j}\circ\theta_k-f_{\alpha,j}\circ\theta\rVert_\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 [54 ] . 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 \lVert\Psi(P_0)\rVert^2=\lim_k Z_k^2C_k(0)=1 ∥ Ψ ( P 0 ) ∥ 2 = lim k Z k 2 C k ( 0 ) = 1 ⟨ Ω , Ψ ( P 0 ) ⟩ = S 1 ( f 0 ) = 0 \langle\Omega,\Psi(P_0)\rangle=S_1(f_0)=0 ⟨ Ω , Ψ ( P 0 )⟩ = S 1 ( f 0 ) = 0
(iv) For a polynomial P P P τ t P \tau_tP τ t P t ≥ 0 t\ge0 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 t P ) ⟩ q_P(t)=\lVert\Psi(\tau_tP)\rVert^2=\langle\Psi(P),\Psi(\tau_{2t}P)\rangle q P ( t ) = ∥ Ψ ( τ t P ) ∥ 2 = ⟨ Ψ ( P ) , Ψ ( τ 2 t 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)\le 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)\le 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 ) 1 − 2 − m q P ( 2 m t ) 2 − m q_P(t)\le q_P(0)^{1-2^{-m}}q_P(2^mt)^{2^{-m}} q P ( t ) ≤ q P ( 0 ) 1 − 2 − m q P ( 2 m t ) 2 − m m m m m → ∞ m\to\infty m → ∞ q P ( t ) ≤ q P ( 0 ) q_P(t)\le 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\ge0 T ( t ) = T ( t /2 ) 2 ≥ 0 ∥ Ψ ( τ t P ) − Ψ ( P ) ∥ → 0 \lVert\Psi(\tau_tP)-\Psi(P)\rVert\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\ge0 H ≥ 0 [54 ] . Spatial translations preserve the form, have the opposite translations as inverses, commute with τ t \tau_t τ t f 0 , t = f 0 ⋅ ( ⋅ − t ) f_{0,t}=f_0\mathord\cdot(\mathord\cdot-t) f 0 , t = f 0 ⋅ ( ⋅ − t ) 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 0 ‾ ∘ θ k ) A k ( f 0 , t − f 0 ) ) , C_k(t)-C_k(0)=\omega_k\left(A_k(\overline{f_0}\circ\theta_k)A_k(f_{0,t}-f_0)\right), C k ( t ) − C k ( 0 ) = ω k ( A k ( f 0 ∘ θ k ) A k ( f 0 , t − f 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\lVert f_{0,t}-f_0\rVert_\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\lVert f_0\rVert_\infty L 0 = ( K 2 / c ) sup k ∣ Q ∣ k ∥ f 0 ∥ ∞ 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\lVert f_{0,\sigma}-f_0\rVert_\infty\le\frac12 L 0 ∥ f 0 , σ − f 0 ∥ ∞ ≤ 2 1 C k ( σ ) / C k ( 0 ) ≥ 1 2 C_k(\sigma)/C_k(0)\ge\frac12 C k ( σ ) / C k ( 0 ) ≥ 2 1 Theorem 47 (iv) applies.
(v) For n ≥ 3 n\ge3 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 k n / 2 − 1 B_k^{n/2-1} B k n /2 − 1 n ≥ 3 n\ge3 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 ν k = Ψ P k − ω k ( P k ) Ω k \nu_k=\Psi_{P_k}-\omega_k(P_k)\Omega_k ν k = Ψ P k − ω k ( P k ) Ω k t t t k k k
ω k ( ϑ ( P k ‾ ) τ t / a k P k ) − ∣ ω k ( P k ) ∣ 2 = ⟨ ν k , T k t / a k ν k ⟩ ≤ e − m k t ∥ ν k ∥ 2 , \omega_k\left(\vartheta(\overline{P_k})\frac{\tau_t^{/a_k}}{}P_k\right)-|\omega_k(P_k)|^2
=\langle\nu_k,T_k^{t/a_k}\nu_k\rangle\le e^{-m_k t}\lVert\nu_k\rVert^2, ω k ( ϑ ( P k ) τ t / a k P k ) − ∣ ω k ( P k ) ∣ 2 = ⟨ ν k , T k t / a k ν k ⟩ ≤ e − m k t ∥ ν k ∥ 2 , and both sides are finite combinations of moments, which converge by (ii) and the approximation of θ \theta θ ⟨ ν , e − t H ν ⟩ ≤ e − m ∗ t ∥ ν ∥ 2 \langle\nu,e^{-tH}\nu\rangle\le e^{-m_*t}\lVert\nu\rVert^2 ⟨ ν , e − t H ν ⟩ ≤ e − m ∗ t ∥ ν ∥ 2 ν = Ψ ( P ) − ⟨ Ω , Ψ ( P ) ⟩ Ω \nu=\Psi(P)-\langle\Omega,\Psi(P)\rangle\Omega ν = Ψ ( P ) − ⟨ Ω , Ψ ( P )⟩ Ω Ω ⊥ \Omega^\perp Ω ⊥ Ω ⊥ \Omega^\perp Ω ⊥ t ≥ 0 t\ge0 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 47 (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 56. The amplitude scale cannot be dropped for a bounded field. Suppose Var ϕ k ( 0 ) → 0 \operatorname{Var}\phi_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)\le\omega_k(|A_k|^2)\le\left(a_k^4\sum_x|f_0(a_kx)|\right)^2\operatorname{Var}\phi_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}\phi_k(0) Var ϕ k ( 0 ) r k r_k r k
Remark 57. Let 0 < ϵ < 1 0<\epsilon<1 0 < ϵ < 1 m > 0 m>0 m > 0 π \pi π { − 1 , 0 , 1 } \{-1,0,1\} { − 1 , 0 , 1 } ϵ / 2 \epsilon/2 ϵ /2 1 − ϵ 1-\epsilon 1 − ϵ ϵ / 2 \epsilon/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\ge0 P t ≥ 0 m m m X ( s ) = s / ϵ X(s)=s/\sqrt{\epsilon} 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/\epsilon 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 \epsilon ϵ t 1 ≤ t 2 ≤ t 3 ≤ t 4 t_1\le t_2\le t_3\le 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}(X P_{t_3-t_2}X^2))=q_1q_3(q_2/\epsilon+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/\epsilon-q_2-2q_2^2) q 1 q 3 ( q 2 / ϵ − q 2 − 2 q 2 2 ) ϵ → 0 \epsilon\to0 ϵ → 0 ϵ \epsilon 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 ∥ \|T_kv\|\le e^{-m_k}\|v\| ∥ 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 58. (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)\le 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 47 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\ge 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\le a_k\log2/\sigma m k ≤ a k log 2/ σ k k k σ \sigma σ Theorem 55 (iv), which depends only on 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\le m_k\le a_k\log2/\sigma m ∗ a k ≤ m k ≤ a k log 2/ σ m ∗ ≤ log 2 / σ m_*\le\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\|f_0\|_\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|\ge R_k}}w_k(z)\ge\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\ge0 u k ≥ 0 Z 3 \mathbb{Z}^3 Z 3 μ k R k ≤ log ( 2 / γ ) \mu_kR_k\le\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 47 , 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 σ > 0 \sigma>0 σ > 0 Theorem 47 (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 46 (iv)).
(ii) Theorem 55 (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)\ge\frac12 C_k(0) C k ( σ ) ≥ 2 1 C k ( 0 ) for large k k k e − m k σ / a k ≥ 1 2 e^{-m_k\sigma/a_k}\ge\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\ge\log(1/\alpha)/(2n_k)\ge 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 )) Theorem 55 (iv). A point x x x a k x 0 ≤ − t 0 + a k a_kx_0\le-t_0+a_k a k x 0 ≤ − t 0 + a k y y y a k y 0 ≥ t 0 a_ky_0\ge t_0 a k y 0 ≥ t 0 ∣ x 0 − y 0 ∣ ≥ R k |x_0-y_0|\ge 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 ∑ a k y ∈ Q ∑ 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 ) , C_k(0)\le K_2r_k^2\|f_0\|_\infty^2a_k^8\sum_{a_ky\in Q}\sum_{\substack{x_0-y_0\ge R_k}}w_k(x-y)
\le K_2r_k^2\|f_0\|_\infty^2|Q|_ka_k^4\sum_{|z_0|\ge R_k}w_k(z), C k ( 0 ) ≤ K 2 r k 2 ∥ f 0 ∥ ∞ 2 a k 8 a k y ∈ Q ∑ 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 c\,r_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'\le0 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|\ge R'}e^{-\mu_k|j|}}{\sum_{j\in\mathbb{Z}}e^{-\mu_k|j|}}
=\frac{2e^{-\mu_kR'}}{1+e^{-\mu_k}}\le2e^{-\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}\ge\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 [56 ] , 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 Theorem 55 (v) the limit can be non-Gaussian only if 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)\ge2/\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 55 records: for a smooth f 0 f_0 f 0 μ k \mu_k μ k [ 0 , ∞ ] [0,\infty] [ 0 , ∞ ] Theorem 47 , 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 59. 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 , ( s , 0 ) ) f_0(\cdot-s,(s,0)) f 0 ( ⋅ − s , ( 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 ∥ ∂ 0 m f 0 ∥ ∞ 2 B_m=(K_2/c)\sup_k|Q|_k\|\partial_0^mf_0\|_\infty^2 B m = ( K 2 / c ) sup k ∣ Q ∣ k ∥ ∂ 0 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_ma_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 ) Theorem 55 (iii)–(iv) lies in the domain of 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 − 2 B 1 σ C k ( 0 ) C_k(\sigma)\ge e^{-2\sqrt{B_1}\sigma}C_k(0) C k ( σ ) ≥ e − 2 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 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(\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 A k T_k^j\Psi_{A_k}=\Psi_{A_k} T k j Ψ A k = Ψ A k τ j A k = A k ( f 0 ( ⋅ − ( j a k , 0 ) ) ) \tau_jA_k=A_k(f_0(\cdot-(ja_k,0))) τ j A k = A k ( f 0 ( ⋅ − ( j a k , 0 ))) ( 1 − T k ) m Ψ k = Z k Ψ ( D k f 0 ) (1-T_k)^m\Psi_k=Z_k\Psi(D_kf_0) ( 1 − T k ) m Ψ k = Z k Ψ ( D k f 0 ) Theorem 55 (iv) its squared norm is the covariance 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 ∥ ∂ 0 m f 0 ∥ ∞ \|D_kf_0\|_\infty\le a_k^m\|\partial_0^mf_0\|_\infty ∥ D k f 0 ∥ ∞ ≤ a k m ∥ ∂ 0 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 \int(1-e^{-a_kE})^{2m}\,d\mu_k ∫ ( 1 − e − a k E ) 2 m d μ k 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
Proposition 59. (continued) and hence 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
and 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_k C_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 58(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 47 (iv) at every separation. The open input of Theorem 47 is therefore implied by these two bounds for one smooth f 0 f_0 f 0
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_pN^{-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 u T U e U_e\mapsto e^{uT}U_e U e ↦ e u T 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 60. (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 2 q / 2 \mathbb{E}_{\omega}E_e^m\le m!(24/N)^m2^{q/2} E ω E e m ≤ m ! ( 24/ N ) m 2 q /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}\langle\nabla_eS_e,\nabla_eF\rangle. E ω [ W e F ] = − E ω ⟨ ∇ e S e , ∇ e F ⟩ . In particular 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) Let ω \omega ω A = ∑ e h e W e A=\sum_e h_eW_e A = ∑ e h e W e 2 2 2 ω ( ϑ ( A ‾ ) A ) = 0 \omega(\vartheta(\overline{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 − 3 t 2 / N b(S_e(e^{t/\sqrt b}U_e)-S_e(U_e))\ge t\sqrt bX_TS_e-3t^2/N b ( S e ( e t / b U e ) − S e ( U e )) ≥ t b X T S e − 3 t 2 / N e b S e / Z e^{bS_e}/Z e b S 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 0 e 2 η ⟨ G 0 , Y ⟩ e^{\eta\lvert Y\rvert^2}=\mathbb{E}_{G_0}e^{\sqrt{2\eta}\langle G_0,Y\rangle} e η ∣ Y ∣ 2 = E G 0 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 6 η ∣ G 0 ∣ 2 / N = ( 1 − 12 η / N ) − q / 2 \mathbb{E}e^{6\eta\lvert G_0\rvert^2/N}=(1-12\eta/N)^{-q/2} E e 6 η ∣ G 0 ∣ 2 / N = ( 1 − 12 η / N ) − q /2 η = N / 24 \eta=N/24 η = N /24 x m ≤ m ! η − m e η x x^m\le m!\,\eta^{-m}e^{\eta x} x m ≤ m ! η − m e η 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 W f ∇ e S e e^{bS_e}W_f\nabla_eS_e e b S e W f ∇ e S e 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 1 1 1 ϑ ( A ‾ ) \vartheta(\overline A) ϑ ( A ) 0 0 0 e e e ω ( ϑ ( A ‾ ) W e ) = 0 \omega(\vartheta(\overline A)W_e)=0 ω ( ϑ ( A ) W e ) = 0 e e e
Proposition 61. 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(\frac12\operatorname{Tr}Q_p\right)\ge\frac{1}{2^{19}(1+6b)^2},\qquad
\mathbb{E}_{\omega}\left[1-\frac12\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 \frac12\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^2e^{-nt} e − r t ≤ e 2 e − n t e 2 ∫ 0 2 e − n t d t ≤ e 2 / n e^2\int_0^{2}e^{-nt}dt\le e^2/n e 2 ∫ 0 2 e − n t d t ≤ e 2 / n I = [ 1 / ( 4 n ) , 1 / ( 2 n ) ] I=[1/(4n),1/(2n)] I = [ 1/ ( 4 n ) , 1/ ( 2 n )] 1 2 n − 1 / 2 e − 1 / 2 \frac12n^{-1/2}e^{-1/2} 2 1 n − 1/2 e − 1/2 J = [ 1 / n , 3 / ( 2 n ) ] J=[1/n,3/(2n)] J = [ 1/ n , 3/ ( 2 n )] 1 2 n − 1 / 2 e − 3 / 2 \frac12n^{-1/2}e^{-3/2} 2 1 n − 1/2 e − 3/2 P ( T ∈ I ) ≥ 1 / 24 P(T\in I)\ge1/24 P ( T ∈ I ) ≥ 1/24 P ( T ∈ J ) ≥ 1 / 336 P(T\in J)\ge1/336 P ( T ∈ J ) ≥ 1/336 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=\frac12\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)\ge3/(8n) 1 − X 2 = T ( 2 − T ) ≥ 3/ ( 8 n ) E ( 1 − X 2 ) / 3 ≥ 1 / ( 192 n ) ≥ 2 − 19 n − 2 \mathbb{E}(1-X^2)/3\ge1/(192n)\ge2^{-19}n^{-2} E ( 1 − X 2 ) /3 ≥ 1/ ( 192 n ) ≥ 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 ) 3 / 2 ) \frac{d}{dx}(e^{rx}(1-x^2)^{3/2}) d x d ( e r x ( 1 − x 2 ) 3/2 ) [ − 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 ) EX \leq\left(\sqrt{9+4r^2}-3\right)/(2r) \leq2r/(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 ) E[1-x\cdot v]\geq1-EX\geq3/(2r+3)\geq1/(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\geq4 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 62. 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 ) , \begin{aligned}
\left\|\operatorname{Law}_{\omega}\left((V_e)_{e\in\mathcal{E}}\right)-m^{\otimes K}\right\|_{\mathrm{TV}}\leq\frac{K}{2}e^{3b}\left(\frac{4b}{4b+1}\right)^{B-3} \\
&\leq\frac{K}{2}\exp\left(3b-\frac{B-3}{4b+1}\right),
\end{aligned} 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|\leq1 ∣ F ∣ ≤ 1 ϵ \epsilon ϵ 0 ≤ ω ( ϑ ( F c ‾ ) F c ) ≤ 6 ϵ 0\leq\omega(\vartheta(\overline{F^c})F^c)\leq6\epsilon 0 ≤ ω ( ϑ ( F c ) 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\leq r\leq B-2. No plaquette contains two selected links: collinear links do not share a plaquette, parallel coarse edges are 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\leq r_i\leq6b 0 ≤ r i ≤ 6 b l ≥ 1 l\geq1 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\geq0 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)=EX\leq4b/(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\geq1 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 \frac{1}{2}e^{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 c ‾ ) F c ) \omega(\vartheta(\overline{F^c})F^c) ω ( ϑ ( F c ) F c ) 2 ϵ 2\epsilon 2 ϵ ∣ ω ( F ) ∣ 2 |\omega(F)|^2 ∣ ω ( F ) ∣ 2 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 Theorem 47 are not bounded functions of bare coarse links, and none of the three propositions bears on the slice ratio, the tree bounds or (D).
The two residues
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 ≤ ℓ n\leq\ell n ≤ 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 [40 ] , 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 Balaban’s renormalisation group analysis [4 , 5 , 6 ] gives the ultraviolet part in finite volume.
Remark 63. 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 ξ \xi ξ K c K_c K c ξ 3 \xi^3 ξ 3 D ( n , α ) D(n,\alpha) D ( n , α ) n ≳ ξ / α n\gtrsim\xi/\alpha n ≳ ξ / α ℓ ≳ ξ a / α \ell\gtrsim\xi a/\alpha ℓ ≳ ξ a / α ξ \xi ξ
The count of Remark 63 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 64. Let d ≥ 1 d\geq1 d ≥ 1 b > 0 b>0 b > 0 t ≥ 0 t\geq0 t ≥ 0 n ≥ 2 n\geq2 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(\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\}\subset V}( \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}^{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
(ii) 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)\geq(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\leq na\leq\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\geq0 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 ≤ ∣ Δ ∣ ∑ x ( G B j ) x W_1^d\leq|\Delta|\sum_x(GB_j)_x W 1 d ≤ ∣Δ∣ ∑ x ( G B j ) x 1 1 1 ∑ 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}G L_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\leq 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}\geq(\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)\geq\left(\frac{s_1^2}{ns_2}\right)^d\frac{t}{t+2ds_1/s_2}\geq\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)\geq5/6 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)\leq10d/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\geq(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\geq3 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 64(i).
Proposition 65. Let b > 0 b>0 b > 0 n n n V V V φ \varphi φ Z d \mathbb{Z}^d Z d 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 V ( ⋅ ∣ η ) \gamma_V(\cdot\mid\eta) γ V ( ⋅ ∣ 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\neq\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 , η ′ ) = ∣ η − η ′ ∣ \delta(\eta,\eta')=|\eta-\eta'| δ ( η , η ′ ) = ∣ η − η ′ ∣
∑ c K c ≥ ∣ V ∣ = 48 N n , \sum_c K_c\geq|V|=48N_n, c ∑ K c ≥ ∣ V ∣ = 48 N n , so K c ≥ N n K_c\geq 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 law. The quadratic form ∑ p ∩ V ≠ ∅ ( d φ ) p 2 \sum_{p\cap V\neq\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 ) F F F γ V ( ⋅ ∣ η ) \gamma_V(\cdot\mid\eta) γ V ( ⋅ ∣ 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 1 1 1 ∑ i ∣ φ i ∣ \sum_i|\varphi_i| ∑ i ∣ φ i ∣ ∥ t ∥ 1 \lVert t\rVert_1 ∥ t ∥ 1 C V , j = ∥ u j ∥ 1 C_{V,j}=\lVert u_j\rVert_1 C V , j = ∥ u j ∥ 1
W 1 d V ( γ V ( ⋅ ∣ η ) , γ V ( ⋅ ∣ η ′ ) ) ≤ ∑ j C V , j ∣ η j ′ − η j ∣ . W_1^{d_V}\bigl(\gamma_V(\cdot\mid\eta),\gamma_V(\cdot\mid\eta')\bigr)\leq\sum_j C_{V,j}|\eta'_j-\eta_j|. W 1 d V ( γ V ( ⋅ ∣ η ) , γ V ( ⋅ ∣ 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)\neq\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 V ( ⋅ ∣ d h ℓ ) \gamma_V(\cdot\mid dh_\ell) γ V ( ⋅ ∣ d h ℓ ) γ V ( ⋅ ∣ 0 ) \gamma_V(\cdot\mid0) γ V ( ⋅ ∣ 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|\leq\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 = 2 ∑ c K c 2|V|\leq2\sum_jC_{V,j}=2\sum_cK_c 2∣ V ∣ ≤ 2 ∑ j C V , j = 2 ∑ c K c
The proof uses only that gauge transformations constant on the lines of F F F n ≤ 8 n\leq8 n ≤ 8 1.5 1.5 1.5 11 11 11 N n N_n N n n n n N n N_n N n
Remark 66. The other routes we tried stop at (D) or (E), or earlier.
(a) Disjoint blocks. The block criterion of Lemma 80 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 [14 ] ; rescaling it produces no gap.
(c) Assuming the gap. Hypothesis R of [14 ] 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 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).
What the cube estimate requires
This subsection proves four facts about D ( n , α ) D(n,\alpha) D ( n , α ) n n n Theorem 69 , and Proposition 71 for SU(2)), so that for every fixed n n n β \beta β n n n β \beta β 18
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 67. Suppose that C e e ′ ≤ c 0 C_{e e'}\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_0 b(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 = γ V ( ⋅ ∣ Q ) \mu=\gamma_V(\mathord\cdot\mid Q) μ = γ V ( ⋅ ∣ Q ) ν = γ V ( ⋅ ∣ Q ′ ) \nu=\gamma_V(\mathord\cdot\mid Q') ν = γ V ( ⋅ ∣ Q ′ ) μ \mu μ ν \nu ν π ↦ ∫ δ ( X i , Y i ) d π \pi\mapsto\int\delta(X_i,Y_i)\,d\pi π ↦ ∫ δ ( X i , Y i ) 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^{F}(\mathord\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^{F}(\mathord\cdot\mid X) γ i F ( ⋅ ∣ X ) γ i F ( ⋅ ∣ Y ) \gamma_i^{F}(\mathord\cdot\mid Y) γ i F ( ⋅ ∣ Y ) [67 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^{F}(\mathord\cdot\mid X),\gamma_i^{F}(\mathord\cdot\mid Y)\bigr)\le\sum_{k\in V,k\ne 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\le c_0\sum_{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 ( 1 − 12 c 0 ) ∑ i ρ i ≤ c 0 b ( V , j ) d (1-12c_0)\sum_i\rho_i\le c_0b(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^{d_V}(\mu,\nu)\le\sum_i\rho_i W 1 d V ( μ , ν ) ≤ ∑ i ρ i d d d □ \square □
Proposition 68. 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)\le10m^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_{e e'}\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 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\ge\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 SU(2) and ∣ 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=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=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\le10 a c ≤ 10 8 − a c + b c ≥ 0 8-a_c+b_c\ge0 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 ) ≥ m 2 ( 8 − a c + b c ) ≥ 0 , \begin{aligned}
(10m^3-2m^2)-(a_cm^3-b_cm^2)=m^2\bigl((10-a_c)m+b_c-2\bigr) \\
&\ge m^2(8-a_c+b_c)\ge0,
\end{aligned} ( 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 , 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\le 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\le\widehat{\alpha}_n K c / N n ≤ α n Theorem 1 gives c 0 = ∣ β ∣ = ∣ β W ∣ / 4 c_0=|\beta|=|\beta_W|/4 c 0 = ∣ β ∣ = ∣ β W ∣/4 □ \square □
At β W = 3 10 \beta_W = \frac{3}{10} β W = 10 3 n = 16 n = 16 n = 16 α ^ 16 = 117 128 \hat{\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 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 69. 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 0010 0010 0010 0011 0011 0011 0100 0100 0100 0110 0110 0110 1001 1001 1001 1011 1011 1011 1100 1100 1100 1101 1101 1101 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 Table III the first case occurs for exactly 32 32 32 48 48 48 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 . W_1^{d_V}\bigl(\gamma_V^1,\gamma_V(\mathord{\cdot}\mid Q^g)\bigr)
\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. 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 . Lower bound. The function f ( Q ) = − ∑ i ∈ V δ ( Q i , Q i g ) f(Q)=-\sum_{i\in V}\delta(Q_i,Q_i^g) f ( Q ) = − ∑ i ∈ V δ ( Q i , Q i g ) 1 1 1 d V d_V d V
∫ f d γ V ( ⋅ ∣ Q g ) = − ∑ i δ ( Q i , 1 ) d γ V 1 = − ∣ V ∣ ε V , \int f\,d\gamma_V(\mathord{\cdot}\mid Q^g)
=-\sum_i\delta(Q_i,1)\,d\gamma_V^1
=-|V|\varepsilon_V, ∫ f d γ V ( ⋅ ∣ Q g ) = − i ∑ δ ( Q i , 1 ) d γ V 1 = − ∣ V ∣ ε V , and by the triangle inequality
∫ 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 . Every class has 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^{d_V}\bigl(\gamma_V^1,\gamma_V(\mathord{\cdot}\mid Q^g)\bigr)
\ge32DN_n-2|V|\varepsilon_V. W 1 d V ( 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 70. 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 n n n 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 ) − 1 , \gamma_V^1(G^V\setminus U)\le e^{-N\beta s}\bigl(|U'|e^{-N\beta s/2}\bigr)^{-1}, γ V 1 ( G V ∖ U ) ≤ e − N β s ( ∣ U ′ ∣ e − N β s /2 ) − 1 , which tends to zero, so ε V ( β ) → 0 \varepsilon_V(\beta)\to0 ε V ( β ) → 0 β → ∞ \beta\to\infty β → ∞ Theorem 69 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 S U ( 2 ) \mathrm{SU}(2) SU ( 2 ) Theorem 69 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 71. 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\mathord{\cdot},\mathord{\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 69 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=\frac{(i+j+k)}{\sqrt{3}},\quad z_2=\frac{(i-j-k)}{\sqrt{3}}, z 1 = 3 ( i + j + k ) , z 2 = 3 ( i − j − k ) , z 3 = ( − i + j − k ) 3 , z 4 = ( − i − j + k ) 3 , z_3=\frac{(-i+j-k)}{\sqrt{3}},\quad z_4=\frac{(-i-j+k)}{\sqrt{3}}, z 3 = 3 ( − i + j − k ) , z 4 = 3 ( − i − j + k ) , 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 ) Q ( x , μ ) g ( x + e μ ) − 1 Q^g_{(x,\mu)}=g(x)Q_{(x,\mu)}g(x+e_\mu)^{-1} Q ( x , μ ) g = g ( x ) Q ( x , μ ) g ( x + e μ ) − 1 Theorem 69 , 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(\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}(\gamma_V^1,\gamma_V(\cdot\mid Q^g))\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(\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 ⟩ 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}(\gamma_V^1,\gamma_V(\cdot\mid Q^g))\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)^2d\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 ∣ ) 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 72. 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}\|\rho(p)-\rho(q)\|_{\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 Definition 42 , and suppose that 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}\left|\int\operatorname{Tr}\rho(Q_i)d\gamma_V^1\right|\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)^2d\gamma_V^1\ge(1-\alpha)\int_G\delta(1,q)^2dq, ∣ 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 : S U ( 2 ) → G \phi:\mathrm{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 ) \phi(z_kz_l^{-1}) ϕ ( z k z l − 1 ) G G G δ ( ϕ ( z k ) , ϕ ( z l ) ) = d ∗ \delta(\phi(z_k),\phi(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 ) \phi(z_kz_l^{-1}) ϕ ( z k z l − 1 ) ker ρ \ker\rho ker ρ G G G ϕ ( S U ( 2 ) ) \phi(\mathrm{SU}(2)) ϕ ( SU ( 2 )) z k z l − 1 z_kz_l^{-1} z k z l − 1 { ± 1 } ker ϕ ⊂ { ± 1 } \{\pm1\}\ker\phi\subset\{\pm1\} { ± 1 } 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)=\phi(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 g ) = d ∗ \delta(1,Q^g)=d_* δ ( 1 , Q 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(\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 = d − 1 ∫ Tr ρ ( Q i ) d γ V 1 c_iI=d^{-1}\int\operatorname{Tr}\rho(Q_i)d\gamma_V^1 c i I = d − 1 ∫ Tr ρ ( Q i ) d γ V 1 γ V ( ⋅ ∣ Q g ) \gamma_V(\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 )) 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)^2dq=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)^2d\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 \|\rho(p)-\rho(q)\|_{\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 69 to 1 2 \frac{1}{2} 2 1 D 2 = 9 D^2=9 D 2 = 9 δ 2 \delta^2 δ 2 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 73. Let G = S U ( 2 ) G=\mathrm{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 , \frac{1}{|V|}\sum_{e\in V}\int\delta(1,Q_e)^2d\gamma_V^1\le\frac{n^2(n+2)(2n+1)L(b)}{b}, ∣ V ∣ 1 e ∈ V ∑ ∫ δ ( 1 , Q e ) 2 d γ V 1 ≤ b n 2 ( n + 2 ) ( 2 n + 1 ) L ( b ) , 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 \epsilon=\pm1 ϵ = ± 1 x ′ = x + s ϵ e σ x'=x+s\epsilon e_\sigma x ′ = x + sϵ e σ σ \sigma σ [ 0 , n ) [0,n) [ 0 , n ) t ≤ m t \le m t ≤ m ϵ ′ \epsilon' 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\epsilon'e_\tau x ′ + e μ + t ϵ ′ e τ y + t ϵ e τ y+t\epsilon 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\epsilon'e_\tau x ′ + e μ + t ϵ ′ e τ e σ e_\sigma e σ y + t ϵ e τ y+t\epsilon e_\tau y + t ϵ e τ F F F y y y y y y e e e x x x 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 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 V V V 1 1 1 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}\le1 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)\le 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
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 74. 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(\frac{b}{N}\sum_p\operatorname{Re}\operatorname{Tr}Q_p) exp ( N b ∑ p Re Tr Q p ) b ≥ b 0 b\ge b_0 b ≥ b 0 δ ( p , q ) = ∥ p − q ∥ H S \delta(p,q)=\|p-q\|_{\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 ) 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\left(\frac{1-e^{-\operatorname{ad}X}}{\operatorname{ad}X}\right)\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 [35 Ch. II, Thm. 1.7] ; in particular j ≤ 1 j\le1 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 ∑ 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|\le 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|\le3\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}}\le3\pi\sqrt N ∥ X ∥ HS ≤ 3 π N γ N Ω N 2 − 1 ( 2 N ) − 1 \gamma_N\Omega_{N^2-1}(2\sqrt N)^{-1} γ N Ω N 2 − 1 ( 2 N ) − 1 Ω d \Omega_d Ω d R d \mathbb{R}^d R d ∥ X ∥ H S ≤ r ≤ 1 \|X\|_{\mathrm{HS}}\le r\le1 ∥ X ∥ HS ≤ r ≤ 1 ∣ θ k ∣ ≤ 1 < π |\theta_k|\le1<\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\le\|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\le(\theta_k^2+\theta_l^2)^{1/2}/\sqrt2\le2^{-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\ge1-x^2/6 sin x / x ≥ 1 − x 2 /6
Dividing by v N v_N v N { q : δ ( 1 , q ) ≤ r } \{q:\delta(1,q)\le 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}\ge\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}\le1 r = ( N /9 b ) 1/2 ≤ 1 I ≥ e − 1 I\ge 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)\le L_N(b) log ( 1/ I ) ≤ L N ( b ) ∫ S d γ l ≤ 2 N ∣ V ∣ L N ( b ) / b \int Sd\gamma^l\le2N|V|L_N(b)/b ∫ S d γ l ≤ 2 N ∣ V ∣ L N ( b ) / b
Conclusion. Proposition 72 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)\le 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)\le5n^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
Remark 75. 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\le 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 1 1 1 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 [47 ] 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 76. Let G = S U ( 2 ) G=\mathrm{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\ge\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)\ge\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|\le6b 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}\le1 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)\ge 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)\ge 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\ge1 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\le\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\le2\theta^3/(3\pi)\le\pi^2r^3/12 ( θ − sin θ cos θ ) / π ≤ 2 θ 3 / ( 3 π ) ≤ π 2 r 3 /12 r = 2 9 a ′ − 1 / 2 r=\frac{2}{9}a'^{-1/2} r = 9 2 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)\le24e^{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 9 a ′ − 1 / 2 r/2=\frac19a'^{-1/2} r /2 = 9 1 a ′ − 1/2 e ∈ V e\in V e ∈ V ϵ V \epsilon_V ϵ V
Along the continuum tuning Theorem 69 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 S U ( 2 ) \mathrm{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 77. Let G = S U ( 2 ) G=\mathrm{SU}(2) G = SU ( 2 ) b = β W ≥ 1 b=\beta_W\ge1 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 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=Qc_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|\leq4b ∣ h k ∣ ≤ 4 b
The fields. Let a = 1 2 a=\frac{1}{2} a = 2 1 c = 3 2 c=\frac{\sqrt{3}}{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|\leq4ab=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\leq k\leq9 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 ) , \frac{1}{2}\sum_k|v_{k+1}-v_k|^2=\sum_k(1-\langle q_k,q_{k+1}\rangle)-\frac{1}{b}\sum_k\langle h_k,q_k\rangle+\frac{1}{2}\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{b}{2}\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=\frac{1}{2}\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\geq0\geq(q_{10})_4 ( q 0 ) 4 ≥ 0 ≥ ( q 10 ) 4 E ≤ 1 1280 E\leq\frac{1}{1280} E ≤ 1280 1 Δ = ( 20 E ) 1 / 2 ≤ 1 8 \Delta=(20E)^{1/2}\leq\frac18 Δ = ( 20 E ) 1/2 ≤ 8 1 w = v 0 w=v_0 w = v 0 ∣ w − v k ∣ ≤ Δ |w-v_k|\leq\Delta ∣ w − v k ∣ ≤ Δ k k k 0 ≤ w 4 ≤ Δ 0\leq w_4\leq\Delta 0 ≤ w 4 ≤ Δ r k r_k r k ∣ v k + r k ∣ = 1 |v_k+r_k|=1 ∣ v k + r k ∣ = 1 ∣ ∣ w + r k ∣ 2 − 1 ∣ ≤ Δ ( 2 + Δ ) = : η ||w+r_k|^2-1|\leq\Delta(2+\Delta)=:\eta ∣∣ w + r k ∣ 2 − 1∣ ≤ Δ ( 2 + Δ ) =: η 0 ≤ k ≤ 5 0\leq k\leq5 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|\leq\eta/(2a)=\eta ∣ w l ∣ ≤ η / ( 2 a ) = η l = 1 , 2 , 3 l=1,2,3 l = 1 , 2 , 3 ∣ w 1 ∣ 2 ≥ 3 4 − η |w_1|^2\geq\frac34-\eta ∣ w 1 ∣ 2 ≥ 4 3 − η 3 4 ≤ 3 η 2 + Δ 2 + η ≤ 3 ( 3 64 ) 2 + 1 64 + 3 64 = 2019 4096 \frac34\leq3\eta^2+\Delta^2+\eta\leq3\left(\frac{3}{64}\right)^2+\frac{1}{64}+\frac{3}{64}=\frac{2019}{4096} 4 3 ≤ 3 η 2 + Δ 2 + η ≤ 3 ( 64 3 ) 2 + 64 1 + 64 3 = 4096 2019 K b ( x , y ) ≤ e − b / 1280 K_b(x,y)\leq e^{-b/1280} K b ( x , y ) ≤ e − b /1280 x 4 ≥ 0 ≥ y 4 x_4\geq0\geq y_4 x 4 ≥ 0 ≥ y 4
The top eigenvalue. Let ϵ = b − 1 / 2 \epsilon=b^{-1/2} ϵ = b − 1/2 ϵ ≤ 1 \epsilon\leq1 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 24 \frac{2}{\pi}\int_0^\epsilon\sin^2\theta\,d\theta\geq\frac{8\epsilon^3}{3\pi^2}>\frac{\epsilon^3}{24} π 2 ∫ 0 ϵ sin 2 θ d θ ≥ 3 π 2 8 ϵ 3 > 24 ϵ 3 q k q_k q k ϵ \epsilon 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|\leq\epsilon ∣ v k − c e 4 ∣ ≤ ϵ E ≤ 20 ϵ 2 E\leq20\epsilon^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 / 24 ) 10 = e − 2024 − 10 b − 15 \Lambda_b\geq e^{-20}(\epsilon^3/24)^{10}=e^{-2024^{-10}b^{-15}} Λ b ≥ e − 20 ( ϵ 3 /24 ) 10 = e − 202 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\geq y_4}\psi_b(x)K_b(x,y)\psi_b(y)\,dx\,dy \\
&\leq 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\leq\frac12 ∫ x 4 > 0 ψ b ≤ 2 1 Λ b , 1 ≥ ⟨ ϕ , K b ϕ ⟩ \Lambda_{b,1}\geq\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 Proposition 77 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
Acknowledgements
This paper rests on the work of many earlier scientists. Lattice gauge theory is due to Wilson [69 ] and its Hamiltonian form to Kogut and Susskind [44 ] ; reflection positivity and the reconstruction of the transfer matrix to Osterwalder and Schrader [54 , 55 ] , and their lattice form, with the first proof of a gap at strong coupling, to Osterwalder and Seiler [56 , 60 ] . The uniqueness method is due to Dobrushin’s [26 , 27 , 28 ] , the curvature method is that of Bakry and Émery [2 , 3 ] in the form of Shen, Zhu and Zhu [61 ] , the gauge fixing on a forest is Creutz’s [24 ] , and the curvature of Lie groups is Milnor’s [52 ] . The analysis of the single-link law uses results of Barta [8 ] , Biernacki and Krzyż [18 ] , McShane [51 ] , Stein and Weiss [64 ] and Watson [68 ] , and the continuum problem is the one formulated by Jaffe and Witten [40 ] and studied by Balaban [4 , 5 , 6 ] . The Yang–Mills manuscripts of Sussmann and collaborators [65 , 66 ] were read in the preparation of Sec. IX.
Data and code availability
The analytic proofs are complete as written, except the evaluations in Propositions Propositions 34 and 35 and Lemma Lemma 38 and the numerical bounds of Lemma Lemma 16, which are computations with rigorous error bounds, and the finite check of Lemma Lemma 12. The repository creelie/4Y-M contains the \LaTeX{} source of this paper; the programs of that computation, in Python with python-flint (FLINT and Arb) and a C kernel for products of Taylor polynomials, with their outputs and the list of certified boxes; the programs and outputs of the earlier certificate at β W = 45 64 \beta_W=\frac{45}{64} β W = 64 45 § 2.4 H, with Monte Carlo estimates of the influence coefficients and of ε V \varepsilon_V ε V [11 ] , release v3.0.5, as described in the footnote to that theorem; release v3.1.0 [12 ] records the engine’s Yang–Mills runs in its failure log, whose last Yang–Mills entry is the change from single links to blocks of several links; the plaquette blocks of § 1.4 , the bound of Lemma 15, their SU(3) version in Sec. VIII and the certified constants were found and checked by the authors. 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 ′ § 1.2 , 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 78. 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^F_e 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^F_e(\cdot\mid\sigma) γ e F ( ⋅ ∣ σ ) γ e F ( ⋅ ∣ σ ′ ) \gamma^F_e(\cdot\mid\sigma') γ e F ( ⋅ ∣ σ ′ ) [67 Cor. 5.22] . Both μ i \mu_i μ i γ e F \gamma^F_e 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'}\neq0 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 □ \square □
Corollary 79. 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 [43 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^F_e 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\leq e^{-\lambda\operatorname{dist}(S_F,S_H)} w z ≤ e − λ dist ( S F , S H ) z ∈ S H z\in S_H z ∈ 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 ) \lvert\mu^\sigma(F)-\mu^{\sigma'}(F)\rvert\leq L_F\sum_{b\in S_F}\rho(b)\leq 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 ′ ) □ \square □
Lemma 80. Let the blocks partition A A A C B j C_{Bj} C B j § 1.4 , with ∑ B C B j ≤ α < 1 \sum_B C_{Bj}\leq\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 2 2 2 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 \gamma^{\mathcal{B}} γ B 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 , and 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_{B j}\rho(j) ∑ j C B j ρ ( j ) w ˉ B = max i ∈ B w i \bar w_B=\max_{i\in B}w_i w ˉ B = max i ∈ B w i
∑ i ∈ B w i ρ ( i ) ≤ w ˉ B ∑ j C B j ρ ( j ) , \sum_{i\in B}w_i\rho(i)\le\bar w_B\sum_j C_{B j}\rho(j), i ∈ B ∑ w i ρ ( i ) ≤ w ˉ B j ∑ C B j ρ ( j ) , and w ˉ B ≤ e 2 λ w j \bar w_B\le e^{2\lambda}w_j w ˉ B ≤ e 2 λ w j C B j ≠ 0 C_{B j}\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 78.
Lemma 81. 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^{\mathcal 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^{\mathcal F}(fg)=g\gamma_e^{\mathcal F}f 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^{\mathcal F}f)-\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^{\mathcal 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^{\mathcal F} γ e F P η P_\eta P η γ e F \gamma_e^{\mathcal 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 ˉ x = max i ∈ V x w i \bar w_x=\max_{i\in V_x}w_i w ˉ x = max i ∈ V x w i
∑ i ∈ V x w i ρ ( i ) ≤ w ˉ x ∑ j ∉ V x C V x , j ρ ( j ) , \sum_{i\in V_x}w_i\rho(i)\le\bar w_x\sum_{j\notin V_x}C_{V_x,j}\rho(j), i ∈ V x ∑ w i ρ ( i ) ≤ w ˉ x j ∈ / V x ∑ C V x , j ρ ( j ) , and w ˉ x ≤ e λ ρ n w j \bar w_x\le e^{\lambda\rho_n}w_j w ˉ 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_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}\le Kc(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 ∪ 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\cup 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 ∪ 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}\left(1+c_ne^{\lambda\rho_n}\right)\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 79 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 = ∅
The plaquette blocks
Table IV 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 IV. 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
The program cover.py in the folder strong-coupling/gauge-fixed/blocks/ of the archive finds this set by an exact-cover search on the torus of side two and checks Lemma 12 on the tori of side four and six. The program constants.py evaluates the bounds of Lemma 16 in ball arithmetic with python-flint at 120 bits: the series of Z 0 Z_0 Z 0 s = r 2 s=r^2 s = r 2 s ≤ 5 s\le5 s ≤ 5 ( 5 / 4 ) 40 40 ! / ( 40 ! 41 ! ) (5/4)^{40}40!/ (40!41!) ( 5/4 ) 40 40 ! / ( 40 ! 41 !) alpha.py evaluates the constants of (3) and α \alpha α β W \beta_W β W
For SU(3) the folder strong-coupling/gauge-fixed/blocks/su3/ contains constants.py, which evaluates the bounds of Lemma 38 on the 841 boxes of side 3 16 \frac{3}{16} 16 3 r = 3 2 r=\frac{3}{2} r = 2 3 19 100 \frac{19}{100} 100 19 r = 38 25 r=\frac{38}{25} r = 25 38 alpha3.py, which reads these bounds and the cover of Proposition 34 or Proposition 35 and evaluates the constants of (15) and α 3 \alpha_3 α 3 β W = 3 4 \beta_W=\frac{3}{4} β W = 4 3 --r 38/25 --thr 1.03.
The inequality is not tight. In floating point the least eigenvalue of the whole form over W ∈ U ( N ) W\in\mathrm{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 blocks of Lemma 20
Diagonal blocks. Let w a = e i ϕ w_a=e^{i\phi} w a = e i ϕ f p = − cos ϕ ( y 1 + y 2 − y 3 − y 4 ) 2 f_p=-\cos\phi\left(y_1+y_2-y_3-y_4\right)^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\left(y_1y_4-y_1y_2+y_2y_3-y_3y_4\right) 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\left(y_1+y_2-y_3-y_4\right)^2-\frac{9}{5}\left(y_1y_4-y_1y_2+y_2y_3-y_3y_4\right) 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 certified computation
Truncation. For a corner b b b Z Z Z ( b , b ‾ ) (b,\overline{b}) ( b , b ) E H [ A t ( U D ) B t ( U D ′ ) ] \mathbb{E}_{H}[A_t(UD)B_t(UD^{\prime})] E H [ A t ( U D ) B t ( U D ′ )] A t A_t A t ∑ λ c λ s λ ( U D ) \sum_\lambda c_\lambda s_\lambda(UD) ∑ 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 ∥ n n ! T(\lVert D\rVert)=\sum_{n>34}\frac{\lVert D\rVert^n}{n!} T (∥ D ∥) = n > 34 ∑ n ! ∥ D ∥ n in modulus. On the polydisc ∣ 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 ∣ ρ − ∣ α ∣ γ 2^{|\alpha|}\rho^{-|\alpha|}\gamma 2 ∣ α ∣ ρ − ∣ α ∣ γ γ = ∑ k ∣ b k ∣ \gamma=\sum_k|b_k| γ = ∑ k ∣ b k ∣ ∣ Tr ( U D ) ∣ ≤ ∑ k ∣ b k ∣ |\operatorname{Tr}(UD)|\leq\sum_k|b_k| ∣ Tr ( U D ) ∣ ≤ ∑ 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 + ( e γ / 2 + ε α ) ε β \varepsilon_\alpha e^{\gamma/2}+ (e^{\gamma/2}+\varepsilon_\alpha)\varepsilon_\beta ε α e γ /2 + ( e γ /2 + ε α ) ε β 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 ′ X=DD^{\prime} X = D D ′ det D \det D det D det D ′ \det D^{\prime} det D ′
Rounding. The truncated series is a Taylor polynomial of total degree six in the 36 real directions of ( D − b , D ′ − b ‾ ) (D-b,D^{\prime}-\overline{b}) ( D − b , D ′ − b ) n n n η n = n u / ( 1 − n u ) \eta_n=nu/(1-nu) η n = n u / ( 1 − n u ) u = 2 − 53 u=2^{-53} u = 2 − 53 [36 Sec. 3.1] . With C C C 2 − 30 Z 2^{-30}\mathbb{Z} 2 − 30 Z 2 31 q h 2^{31}q_h 2 31 q h X ( 2 31 q h ) X(2^{31}q_h) X ( 2 31 q h ) 12 ⋅ 2 31 ( ϕ h + Δ q h ) 12\cdot2^{31}(\phi_h+\Delta q_h) 12 ⋅ 2 31 ( ϕ h + Δ q h ) 12 ⋅ 2 31 ⟨ ∇ ϕ k , ∇ q h ⟩ 12\cdot2^{31}\langle\nabla\phi_k,\nabla q_h\rangle 12 ⋅ 2 31 ⟨ ∇ ϕ k , ∇ q h ⟩ h h h Z [ i ] \mathbb{Z}[i] Z [ i ] 2 50 2^{50} 2 50 Z Z Z [41 ] .
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 / 2 \frac{1}{2}H\delta^2/2 2 1 H δ 2 /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
Running time. The moments at one corner take about one and a half seconds and the bound on one box about two and a half seconds on one core. The cover runs in about an hour on four cores and writes every certified box with its θ \theta θ
The transfer matrix
This appendix recalls the Osterwalder–Schrader reconstruction for the lattice [54 , 55 , 56 ] in the form used for the gaps, following our earlier paper [13 ] .
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_t H) ⟨ Ψ F , T t Ψ H ⟩ = ω ( ϑ ( F ) τ t H ) ϑ \vartheta ϑ τ t \tau_t τ t 19
Proposition 82. 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_t F)-|\omega(F)|^2 \le C_F e^{-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 = \lVert F\rVert_\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_t F)-|\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}\lVert\Psi'_F\rVert^2] [ 0 , e − m t ∥ Ψ F ′ ∥ 2 ] ∥ Ψ F ∥ 2 ≤ ∥ F ∥ ∞ 2 \lVert\Psi_F\rVert^2\le\lVert F\rVert_\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_F e^{-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 83. 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 [57 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 1 1 1 ν ( { 1 } ) = ∥ P ψ ∥ 2 = ⟨ P ψ , ψ ⟩ = 0 \nu(\{1\})=\lVert P\psi\rVert^2=\langle P\psi,\psi\rangle=0 ν ({ 1 }) = ∥ P ψ ∥ 2 = ⟨ P ψ , ψ ⟩ = 0 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 84. 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 ∥ ∞ \lVert h_j\rVert_\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 85. 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\le\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 , ∞ ] □ \square □
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 Theorem 47 a bound uniform in 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 [20 Chs. 2–4] ).
Lemma 86. 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 \hat c_\lambda=d_\lambda^{-1}\int w\chi_\lambda\,dU c ^ λ = d λ − 1 ∫ w χ λ d U ∫ w ( U ) ρ λ ( U ) d U = c ^ λ I \int w(U)\rho_\lambda(U)\,dU=\hat 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=\hat 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=\hat 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}(\int w(U)\rho_\lambda(U^{-1})dU\,\rho_\lambda(V)) ∫ 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 87. Let G = S U ( N ) G=\mathrm{SU}(N) G = SU ( N ) β > 0 \beta>0 β > 0
w β ( U ) = e β N Re Tr U ∫ e β N Re Tr V d V . w_\beta(U)=\frac{e^{\frac{\beta}{N}\operatorname{Re}\operatorname{Tr}U}}{\int e^{\frac{\beta}{N}\operatorname{Re}\operatorname{Tr}V}\,dV}. w β ( U ) = ∫ e N β Re Tr V d V e N β Re Tr U . Then c ^ λ ( β ) = d λ − 1 ∫ w β χ λ d U \hat 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\hat c_\lambda(\beta)\le1 0 ≤ c ^ λ ( β ) ≤ 1 c ^ λ ( β ) < 1 \hat 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 → ∞ ) , \hat 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 ˉ ) = w β ( U ) w_\beta(\bar U)=w_\beta(U) w β ( U ˉ ) = w β ( U ) χ λ ( U ˉ ) = χ λ ( U ) ‾ \chi_\lambda(\bar U)=\overline{\chi_\lambda(U)} χ λ ( U ˉ ) = 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 \hat 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 \hat 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 \hat 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, Re Tr U ≤ N \operatorname{Re}\operatorname{Tr}U\le N Re Tr 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 Re Tr U χ λ \int e^{\frac{\beta}{N}\operatorname{Re}\operatorname{Tr}U}\chi_\lambda ∫ e N β Re Tr U χ λ ∫ e β N Re Tr U \int e^{\frac{\beta}{N}\operatorname{Re}\operatorname{Tr}U} ∫ e N β Re Tr 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 Re Tr e X = 1 − 1 2 N ∣ X ∣ 2 + r ( X ) , 0 ≤ r ( X ) ≤ C ∣ X ∣ 4 , \frac{1}{N}\operatorname{Re}\operatorname{Tr}e^X=1-\frac{1}{2N}|X|^2+r(X),\qquad0\le r(X)\le C|X|^4, N 1 Re Tr 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 ∣ X ∣ 2 / ( 2 N ) e^{-\beta|X|^2/(2N)} e − β ∣ X ∣ 2 / ( 2 N ) 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=je^{\beta r} h = j e β r
c ^ λ ( β ) = ⟨ d λ − 1 χ λ ( e X ) h ⟩ ⟨ h ⟩ + O ( e − c β ) = 1 + ⟨ 1 2 d λ Tr ρ λ ( X ) 2 h ⟩ ⟨ h ⟩ + ⟨ ( c 3 ( X ) + O ( ∣ X ∣ 4 ) ) h ⟩ ⟨ h ⟩ . \begin{aligned}
\hat c_\lambda(\beta)=\frac{\langle d_\lambda^{-1}\chi_\lambda(e^X)h\rangle}{\langle h\rangle}+O(e^{-c\beta}) \\
&=1+\frac{\left\langle\frac{1}{2d_\lambda}\operatorname{Tr}\rho_\lambda(X)^2h\right\rangle}{\langle h\rangle}
+\frac{\left\langle(c_3(X)+O(|X|^4))h\right\rangle}{\langle h\rangle}.
\end{aligned} c ^ λ ( β ) = ⟨ h ⟩ ⟨ d λ − 1 χ λ ( e X ) h ⟩ + O ( e − c β ) = 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 , \left\langle\operatorname{Tr}\rho_\lambda(X)^2\right\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 , which gives the expansion. For S O ( 3 ) \mathrm{SO}(3) SO ( 3 ) j j j 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 ) 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 ( 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 ′ ∣ Theorems 1 and 3 use, and the exact value at b = 0 b=0 b = 0 Theorem 2 uses. The argument is that of our earlier paper [13 ] ; it is given in full so that the present paper is self-contained.
The von Mises–Fisher law
Identify SU(2) with the unit quaternions, that is with the unit sphere S 3 ⊂ R 4 S^3\subset\mathbb{R}^4 S 3 ⊂ R 4 Re Tr U = 2 x 0 ( U ) \operatorname{Re}\operatorname{Tr}U=2x_0(U) Re Tr U = 2 x 0 ( U ) Re Tr ( U V ) = 2 ⟨ u , v ⟩ \operatorname{Re}\operatorname{Tr}(UV)=2\langle u,v\rangle Re Tr ( 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 Re Tr ( Q e A p ) \operatorname{Re}\operatorname{Tr}(Q_eA_p) Re Tr ( 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 Re Tr ( Q e A p ) = ⟨ x , b ⟩ , b = 4 β ∑ p ∋ e A p ‾ ∈ R 4 , N\beta\sum_{p\ni e}\operatorname{Re}\operatorname{Tr}(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 ∑ Re Tr ( 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 ∣ . (17) \kappa:=|b|\le y:=24|\beta|. \tag*{(17)} κ := ∣ b ∣ ≤ y := 24∣ β ∣. (17) The conditional law of Q e Q_e Q e [20 ]
ν b ( d x ) = Z ( κ ) − 1 e ⟨ x , b ⟩ σ ( d x ) (18) \nu_b(dx)=Z(\kappa)^{-1}e^{\langle x,b\rangle}\,\sigma(dx) \tag*{(18)} ν b ( d x ) = Z ( κ ) − 1 e ⟨ x , b ⟩ σ ( d x ) (18) 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 88. 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\mathop{\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 [8 ] 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 89. 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 ) − 1 A(\kappa)\geq\kappa\left(\frac32+\sqrt{\kappa^2+\frac{25}{4}}\right)^{-1} A ( κ ) ≥ κ ( 2 3 + κ 2 + 4 25 ) − 1
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 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 ν c ( g , ⟨ h , x ⟩ ) , (19) L(c,h)=\sup_g\operatorname{Cov}_{\nu_c}(g,\langle h,x\rangle), \tag*{(19)} L ( c , h ) = g sup Cov ν c ( g , ⟨ h , x ⟩) , (19) the supremum over 1 1 1 g g g δ \delta δ d d t ν c + t h ( g ) = Cov ν c + t h ( g , ⟨ h , x ⟩ ) \frac{d}{dt}\nu_{c+th}(g)=\operatorname{Cov}_{\nu_{c+th}}(g,\langle h,x\rangle) d t d ν c + t h ( g ) = Cov ν 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 ∣ ) . (20) W_1(\nu_b,\nu_{b'})\leq|b-b'|\sup_{c\in[b,b']}L\left(c,\frac{b'-b}{|b'-b|}\right). \tag*{(20)} W 1 ( ν b , ν b ′ ) ≤ ∣ b − b ′ ∣ c ∈ [ b , b ′ ] sup L ( c , ∣ b ′ − b ∣ b ′ − b ) . (20) 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 90. 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 . (21) L(c,h)\leq\left(\frac{1}{3}\frac{A}{\kappa}\right)^{1/2}\leq\left(3(1+\sqrt{\kappa^2+9})\right)^{-1/2}\leq\frac{1}{2\sqrt{3}}. \tag*{(21)} L ( c , h ) ≤ ( 3 1 κ A ) 1/2 ≤ ( 3 ( 1 + κ 2 + 9 ) ) − 1/2 ≤ 2 3 1 . (21) (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 ] \operatorname{Cov}(g,\langle h,x\rangle)=\mathbb{E}\left[(g(x)-g(Rx))\langle h,x\rangle;\ \langle h,x\rangle>0\right] 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 , \leq\mathbb{E}\left[2\langle h,x\rangle^2;\ \langle h,x\rangle>0\right]=\mathbb{E}\langle h,x\rangle^2, ≤ 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 91 (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\frac14, 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\frac14|b-b'| 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 \frac14 4 1
The equivalence is (I4) in one direction and the derivative formula above it in the other; the sharpness is Lemma 90(i). The metric matters.21 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\,d\sigma=0 ∫ u d σ = 0
Q ( u ) = ∑ k ≥ 1 1 k ∥ u k ∥ L 2 ( σ ) 2 . (22) \mathcal{Q}(u)=\sum_{k\geq1}\frac{1}{k}\lVert u_k\rVert_{L^2(\sigma)}^2. \tag*{(22)} Q ( u ) = k ≥ 1 ∑ k 1 ∥ u k ∥ L 2 ( σ ) 2 . (22) 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 92. Let u u u S 3 S^3 S 3 ∫ u d σ = 0 \int u\,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\,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 [51 ] ; 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 22 ∂ 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\,dS=\int_B\nabla G\cdot\nabla\psi\,dy ∫ 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=\frac{\pi^2}{2} ∣ B ∣ = 2 π 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\,d\sigma=\frac{1}{2\pi^2}\int_B\nabla G\cdot\nabla\psi\,dy\leq\frac{\lvert B\rvert^{1/2}}{2\pi^2}\left(\int_B\lvert\nabla\psi\rvert^2\,dy\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\,dS=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\,d\sigma sup g ∫ g u d σ R 4 \mathbb{R}^4 R 4 u d σ u\,d\sigma u d σ 23 ∇ 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=d\nu_c/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}\,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\,d\sigma Cov ν c ( g , ⟨ h , x ⟩) = ∫ g u d σ ∫ u d σ = 0 \int u\,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. (23) d_n(\kappa)=\frac{(n+1)I_{n+1}(\kappa)}{I_1(\kappa)},\qquad n\geq0. \tag*{(23)} d n ( κ ) = I 1 ( κ ) ( n + 1 ) I n + 1 ( κ ) , n ≥ 0. (23) Lemma 93. 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\geq0 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 c x 1 = A \mathbb{E}_{\nu_c}x_1=A E ν c x 1 = A E ν c x 2 = 0 \mathbb{E}_{\nu_c}x_2=0 E ν c x 2 = 0 λ ↦ − λ \lambda\mapsto-\lambda λ ↦ − λ 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_n'U_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=e^{\kappa\langle y(t),x\rangle}/Z(\kappa)=\sum_n d_n(\kappa)U_n(\langle y(t),x\rangle) ρ t = e κ ⟨ y ( t ) , x ⟩ / Z ( κ ) = ∑ n d n ( κ ) U n (⟨ y ( t ) , x ⟩) ∂ 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 . 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. 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]=\frac13(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=\frac13n(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\frac13n(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 92 and 93,
L ( κ , φ ) ≤ 1 2 max ( Q z ( κ ) , Q x ( κ ) ) 1 / 2 , (24) L(\kappa,\varphi)\le\frac12\max\bigl(\mathcal{Q}_z(\kappa),\mathcal{Q}_x(\kappa)\bigr)^{1/2}, \tag*{(24)} L ( κ , φ ) ≤ 2 1 max ( Q z ( κ ) , Q x ( κ ) ) 1/2 , (24) and Theorem 91 follows once both parts are at most 1 4 \frac14 4 1 1 4 \frac14 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\frac12 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 93 reduce to closed forms. Write I j I_j I j I j ( κ ) I_j(\kappa) I j ( κ )
Lemma 94. 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 95. Q x ( κ ) ≤ 1 4 \mathcal{Q}_x(\kappa)\le\frac14 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\frac14 Q x ≤ 4 1
D ( κ ) : = ( 8 + 2 κ 2 ) I 1 2 − κ 2 I 0 2 − κ I 1 ( 2 κ ) ≥ 0. (25) \mathcal{D}(\kappa):=(8+2\kappa^2)I_1^2-\kappa^2I_0^2-\kappa I_1(2\kappa)\ge0. \tag*{(25)} D ( κ ) := ( 8 + 2 κ 2 ) I 1 2 − κ 2 I 0 2 − κ I 1 ( 2 κ ) ≥ 0. (25) 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=\frac1{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 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 A ′ = 1 − A 2 − 3 A / κ A'=1-A^2-3A/\kappa A ′ = 1 − A 2 − 3 A / κ
Proposition 96. 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 + ν κ I ν I_\nu=I_{\nu+1}+\frac{\nu}{\kappa}I_\nu I ν = I ν + 1 + κ ν I ν
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 99 of Appendix K, r n + 1 ≤ r 1 r_{n+1}\le r_1 r n + 1 ≤ r 1 r 1 = A ≤ κ / 4 r_1=A\le\kappa/4 r 1 = A ≤ κ /4 − 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 2 d n 2 = 4 A ′ 2 + S 3 − S 2 − 4 I 1 2 κ 2 I 1 2 , \mathcal{Q}_z\le4A'^2+\frac1{\kappa^2}\sum_{n\ge2}n^2d_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 2 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 1 2 = ( j − 1 ) j I j 2 nd_1^2=(j-1)jI_j^2 n d 1 2 = ( j − 1 ) j 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=\frac2\kappa I_1 I 0 − I 2 = κ 2 I 1 I__1-I_3=\frac4\kappa I_2 give I 0 / I 1 = 2 / κ + A I_0/I_1=2/\kappa+A I 0 / I 1 = 2/ κ + A I 2 / I 1 = A I_2/I_1=A I 2 / I 1 = A S 3 / ( κ 2 I 1 2 ) = 1 4 ( 2 / κ + A ) 2 + 1 4 S_3/(\kappa^2I_1^2)=\frac14(2/\kappa+A)^2+\frac14 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^2I_1^2}\ge\frac{I_0I_1+I_1I_2+I_2I_3}{2\kappa I_1^2}
=\frac1{2\kappa}\left(\frac2\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 ) \mathcal{Q}_z\le\frac14+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 b = κ / ( 3 / 2 + s ) A_b=\kappa/(3/2+s) A b = κ / ( 3/2 + s ) A b ≤ A ≤ κ / 4 A_b\le A\le\kappa/4 A b ≤ 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\ge0 a ≥ 0 A ′ ≥ 0 A'\ge0 A ′ ≥ 0 A ′ = 1 − A 2 − 3 A / κ A'=1-A^2-3A/\kappa A ′ = 1 − A 2 − 3 A / κ κ 2 ≤ 12 \kappa^2\le12 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)\le\frac{\kappa}{2}(\kappa^2-8)\le2\kappa ( a + a ′ ) ( κ 2 − 8 ) ≤ 2 κ ( κ 2 − 8 ) ≤ 2 κ Q z ≤ 1 4 + 1 4 α 2 + T ( A ♭ ) \mathcal{Q}_z\le\frac{1}{4}+\frac{1}{4\alpha^2}+T(A_\flat) Q z ≤ 4 1 + 4 α 2 1 + 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 , (26) 4\alpha^2+T(A_\flat)=\frac{(2s-5)p(s)}{4(2s+3)^4}, \tag*{(26)} 4 α 2 + T ( A ♭ ) = 4 ( 2 s + 3 ) 4 ( 2 s − 5 ) p ( s ) , (26) where
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\le\sqrt{88/3} 0 < κ ≤ 88/3 5 2 < s ≤ 577 6 < 4.004 \frac{5}{2}<s\le\frac{\sqrt{577}}{6}<4.004 2 5 < s ≤ 6 577 < 4.004 s ≥ 5 2 s\ge\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 91
At c = 0 c=0 c = 0 ∣ c ∣ ≥ 88 / 3 |c|\ge\sqrt{88/3} ∣ c ∣ ≥ 88/3 0 < ∣ c ∣ < 88 / 3 0<|c|<\sqrt{88/3} 0 < ∣ c ∣ < 88/3 Q x , Q z ≤ 1 4 \mathcal{Q}_x,\mathcal{Q}_z\le\frac{1}{4} Q x , Q z ≤ 4 1 L ≤ 1 4 L\le\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\ge\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\ge\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\ge\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 [8 ] in the form of a ground state transformation.
The computations of Lemma 88
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. ∗ ∗ T a k e \ell\ge1.** Take ℓ ≥ 1. ∗ ∗ T ak e 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\ge3 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\ge2\sin^{-2}\theta V = ℓ ( ℓ + 1 ) sin − 2 θ ≥ 2 sin − 2 θ ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 − L ℓ ≥ 3 -L_\ell\ge3 − L ℓ ≥ 3 ℓ ≥ 1 \ell\ge1 ℓ ≥ 1
**Degree ℓ = 0. ∗ ∗ L e t \ell=0.** Let ℓ = 0. ∗ ∗ L e t − 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\not\equiv0 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\ge3 3 + 25 6 κ 2 sin 2 θ ≥ 3 h h h λ ≥ 3 \lambda\ge3 λ ≥ 3 ℓ ≥ 1 \ell\ge1 ℓ ≥ 1 − L -L − L 3 3 3 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 ( 1 − x 1 2 ) = 3 E x 1 / κ \mathbb{E}(1-x_1^2)=3\mathbb{E}x_1/\kappa E ( 1 − x 1 2 ) = 3 E x 1 / κ 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 ) 24 {}^{24} 24
The Riccati barriers
By Lemma 89(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}\le0, 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}\ge0. 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 ) 25 0 0 0 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 , 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, 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=\frac{\kappa^2}{s(s+2)^2}\ge0 R = s ( s + 2 ) 2 κ 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\ge0 ( 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\ge0 μ D ≥ 0 A ≥ ϕ − A\ge\phi_- A ≥ ϕ − D = ϕ + − A D=\phi_+-A D = ϕ + − A R = ( u − 3 ) 2 u ( 1 + u ) 2 R=\frac{(u-3)^2}{u(1+u)^2} R = u ( 1 + u ) 2 ( u − 3 ) 2 A ≤ ϕ + A\le\phi_+ A ≤ ϕ +
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-\frac{9}{4}=\kappa^2+4 s 2 − 4 9 = 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-\frac{5}{2})}{2s(\frac{3}{2}+s)^2}\le0, ϕ ♭ ′ − F ( κ , ϕ ♭ ) = − 2 s ( 2 3 + s ) 2 5 ( s − 2 5 ) ≤ 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 ) 26 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=\frac{5(s-5/2)}{2s(3/2+s)^2}\ge0 R = 2 s ( 3/2 + s ) 2 5 ( s − 5/2 ) ≥ 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 ) 0 0 0 ( μ D ) ′ = μ R ≥ 0 (\mu D)'=\mu R\ge0 ( μ D ) ′ = μ R ≥ 0 A ≥ ϕ ♭ A\ge\phi_{\flat} A ≥ ϕ ♭ 3 2 + κ 2 + 25 / 4 < 2 + κ 2 + 4 \frac{3}{2}+\sqrt{\kappa^2+25/4}<2+\sqrt{\kappa^2+4} 2 3 + 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 91. 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\ge0}\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_j e^{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 , (27) I_{j-1}-I_{j+1}=\frac{2j}{\kappa}I_j,\qquad I_j'=I_{j+1}+\frac{j}{\kappa}I_j, \tag*{(27)} I j − 1 − I j + 1 = κ 2 j I j , I j ′ = I j + 1 + κ j I j , (27) all of which are in Watson’s treatise [68 ] , §§3.7, 3.71, 2.22. All series below converge absolutely, because 0 < I j ≤ ( κ / 2 ) j e κ / j ! 0<I_j\le(\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 \frac{4-2}{2}=1 2 4 − 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 [64 ] , Ch. IV, §2.
Lemma 97. e κ t = 2 κ ∑ n ≥ 0 ( n + 1 ) I n + 1 ( κ ) U n ( t ) e^{\kappa t}=\frac{2}{\kappa}\sum_{n\ge0}(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 ( 2 π ( 1 − t 2 ) 1 / 2 d t ) L^2\left(\frac{2}{\pi}(1-t^2)^{1/2}\,dt\right) 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 , \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}, 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|\le n+1 ∣ U n ∣ ≤ n + 1 I n + 1 I_{n+1} I n + 1
Proof of Lemma 94
Let F ( θ ) = e κ cos θ = ∑ j I j e i j θ F(\theta)=e^{\kappa\cos\theta}=\sum_j I_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_j ijI_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 ) , \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),
\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 ) , 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\le0 j ≤ 0 j ≥ 1 j\ge1 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\ge1}j^3I_j^2=\frac{\kappa^2}{4}\sum_{j\ge1}j(I_{j-1}^2+I_{j+1}^2-2I_{j-1}I_{j+1}). 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\ge1}j(I_{j-1}^2+I_{j+1}^2)=I_0^2+\sum_{m\ge1}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\ge1}2mI_m^2=\kappa\sum_{m\ge1}(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=\frac12((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 98. 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)-4^m(m+1)(m+2)\right) 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-\frac{1}{m!(m+1)!} D m + 1 = 8 β m + 2 β m − 1 − α m − m ! ( m + 1 )! 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 ) 2 ( m + 2 ) b Δ , 8\beta_m=\frac{4(2m+1)b}{\Delta},\qquad\alpha_m=\frac{(m+1)^2(m+2)b}{\Delta}, 8 β m = Δ 4 ( 2 m + 1 ) b , α m = Δ ( m + 1 ) 2 ( m + 2 ) b , 2 β m − 1 = 2 m ( m + 1 ) ( m + 2 ) b Δ , 1 m ! ( m + 1 ) ! = 4 m ( m + 1 ) ( m + 2 ) Δ , 2\beta_{m-1}=\frac{2m(m+1)(m+2)b}{\Delta},\qquad\frac{1}{m!(m+1)!}=\frac{4^m(m+1)(m+2)}{\Delta}, 2 β m − 1 = Δ 2 m ( m + 1 ) ( m + 2 ) b , 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 99. 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 [18 ] : 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. 
