Introduction
Bloch’s law predicts that the spontaneous magnetization of a three-dimensional ferromagnet falls below its fully polarized value by an amount proportional to T 3 / 2 T^{3/2} T 3/2
The model and the result
Fix S ∈ { 1 2 , 1 , 3 2 , … } S \in\left\{\frac{1}{2},1,\frac{3}{2},\ldots\right\} S ∈ { 2 1 , 1 , 2 3 , … } H S = C 2 S + 1 \mathcal{H}_S=\mathbb{C}^{2S+1} H S = C 2 S + 1 ( e m ) m = − S S (e_m)_{m=-S}^{S} ( e m ) m = − S S
S z e m = m e m , S + e m = S ( S + 1 ) − m ( m + 1 ) e m + 1 ( m < S ) , S + e S = 0. S^z e_m = m e_m,\qquad S^+e_m=\sqrt{S(S+1)-m(m+1)}\,e_{m+1}\quad(m<S),\qquad S^+e_S=0. S z e m = m e m , S + e m = S ( S + 1 ) − m ( m + 1 ) e m + 1 ( m < S ) , S + e S = 0. Set S − = ( S + ) ∗ S^-=(S^+)^* S − = ( S + ) ∗ S x = ( S + + S − ) / 2 S^x=(S^++S^-)/2 S x = ( S + + S − ) /2 S y = ( S + − S − ) / ( 2 i ) S^y=(S^+-S^-)/(2i) S y = ( S + − S − ) / ( 2 i ) J : Z 3 → [ 0 , ∞ ) J:\mathbb{Z}^3\to[0,\infty) J : Z 3 → [ 0 , ∞ )
J ( 0 ) = 0 , J ( z ) = J ( − z ) , ⟨ z : J ( z ) > 0 ⟩ Z = Z 3 . (1) J(0)=0,\qquad J(z)=J(-z),\qquad\langle z:J(z)>0\rangle_{\mathbb{Z}}=\mathbb{Z}^3. \tag*{(1)} J ( 0 ) = 0 , J ( z ) = J ( − z ) , ⟨ z : J ( z ) > 0 ⟩ Z = Z 3 . (1) On Λ N = [ − N , N ] 3 ∩ Z 3 \Lambda_N=[-N,N]^3\cap\mathbb{Z}^3 Λ N = [ − N , N ] 3 ∩ Z 3
H S , N J = − 1 2 ∑ x , y ∈ Λ N J ( y − x ) S x ⋅ S y , M S , N = ∑ x ∈ Λ N S x z . (2) H^J_{S,N}=-\frac{1}{2}\sum_{x,y\in\Lambda_N}J(y-x)S_x\cdot S_y,\qquad M_{S,N}=\sum_{x\in\Lambda_N}S_x^z. \tag*{(2)} H S , N J = − 2 1 x , y ∈ Λ N ∑ J ( y − x ) S x ⋅ S y , M S , N = x ∈ Λ N ∑ S x z . (2) The sum counts each unordered bond twice; the factor 1 / 2 1/2 1/2 ℏ \hbar ℏ T = β − 1 T=\beta^{-1} T = β − 1
For β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R
p β ( h ) = lim N → ∞ 1 β ∣ Λ N ∣ log Tr e − β ( H S , N J − h M S , N ) , m S , J ( β ) = ∂ h + p β ( 0 ) . (3) p_\beta(h)=\lim_{N\to\infty}\frac{1}{\beta|\Lambda_N|}\log\operatorname{Tr}e^{-\beta(H^J_{S,N}-hM_{S,N})},\qquad m_{S,J}(\beta)=\partial_h^+p_\beta(0). \tag*{(3)} p β ( h ) = N → ∞ lim β ∣ Λ N ∣ 1 log Tr e − β ( H S , N J − h M S , N ) , m S , J ( β ) = ∂ h + p β ( 0 ) . (3) The finite-range pressure limit exists and is convex and S S S π \pi π x x x m S , J ( β ) ∈ [ 0 , S ] m_{S,J}(\beta)\in[0,S] m S , J ( β ) ∈ [ 0 , S ]
Define the positive definite matrix
D S , J = S 2 ∑ z ∈ Z 3 J ( z ) z z T . (4) D_{S,J}=\frac{S}{2}\sum_{z\in\mathbb{Z}^3}J(z)zz^{\mathsf T}. \tag*{(4)} D S , J = 2 S z ∈ Z 3 ∑ J ( z ) z z T . (4) It is the quadratic coefficient of the one-magnon dispersion:
ε S , J ( k ) = S ∑ z J ( z ) ( 1 − cos ( k ⋅ z ) ) = k T D S , J k + O ( ∣ k ∣ 4 ) . \varepsilon_{S,J}(k)=S\sum_z J(z)(1-\cos(k\cdot z))=k^{\mathsf T}D_{S,J}k+O(|k|^4). ε S , J ( k ) = S z ∑ J ( z ) ( 1 − cos ( k ⋅ z )) = k T D S , J k + O ( ∣ k ∣ 4 ) . Positive definiteness follows because the interaction support spans R 3 \mathbb{R}^3 R 3
Theorem 1.1 (Bloch’s law). For every fixed S ∈ { 1 2 , 1 , 3 2 , … } S\in\left\{\frac{1}{2},1,\frac{3}{2},\ldots\right\} S ∈ { 2 1 , 1 , 2 3 , … }
lim β → ∞ β 3 / 2 ( S − m S , J ( β ) ) = ζ ( 3 / 2 ) 8 π 3 / 2 det D S , J . \lim_{\beta\to\infty}\beta^{3/2}\bigl(S-m_{S,J}(\beta)\bigr)=\frac{\zeta(3/2)}{8\pi^{3/2}\sqrt{\det D_{S,J}}}. β → ∞ lim β 3/2 ( S − m S , J ( β ) ) = 8 π 3/2 det D S , J ζ ( 3/2 ) . For J ( ± e j ) = 1 J(\pm e_j)=1 J ( ± e j ) = 1 D S , J = S I D_{S,J}=SI D S , J = S I ζ ( 3 / 2 ) / ( 8 π 3 / 2 S 3 / 2 ) \zeta(3/2)/(8\pi^{3/2}S^{3/2}) ζ ( 3/2 ) / ( 8 π 3/2 S 3/2 ) β , h \beta,h β , h h ↓ 0 h\downarrow0 h ↓ 0 β → ∞ \beta\to\infty β → ∞ S , J S,J S , J
Historical context and dependencies
Bloch derived the T 3 / 2 T^{3/2} T 3/2 [2 ] . The bosonic representation of Holstein and Primakoff [10 ] and Dyson’s analysis of spin-wave interactions [6 , 7 ] developed the low-temperature expansion. In particular, Dyson identified the interaction contribution to the magnetization at order T 4 T^4 T 4 [7 ] , Section 9. The leading coefficient in (1.6) is obtained from the low-temperature density of ideal magnons with quadratic energy k T D S , J k \mathbf{k}^{\mathsf{T}}D_{S,J}\mathbf{k} k T D S , J k (3) .
Rigorous ordering theory distinguishes sharply between dimensions and between classical and quantum models. Mermin and Wagner excluded positive-temperature spontaneous magnetic order for the one- and two-dimensional isotropic Heisenberg models under their finite-range hypotheses [12 ] . Infrared bounds proved symmetry breaking for classical models [9 ] , and Dyson, Lieb, and Simon established ordering for classes of quantum antiferromagnets [8 ] . These developments left the standard three-dimensional quantum ferromagnet as a separate problem, recorded by Lieb [11 ] and still discussed as open in recent accounts [16 , 1 ] . The distinct companion manuscript [15 ] , Theorem 1.1 proves low-temperature magnetized equilibrium states for the nearest-neighbor quantum ferromagnet. We use its finite-set algebra and deterministic lattice geometry, with their precise hypotheses stated at the points of use.
The free energy has a different rigorous history. Conlon and Solovej [4 ] and Tóth [17 ] obtained bounds for spin 1 / 2 1/2 1/2 Z 3 \mathbb{Z}^3 Z 3 [5 ] , Theorem 2.1. A zero-field free-energy estimate does not by itself control the one-sided field derivative that defines spontaneous magnetization. Our proof therefore estimates the observable through a representation that remains uniform under arbitrary extra up-spin constraints, and only then passes to pressure increments.
The probabilistic representation belongs to the random-interchange approach of Tóth [17 ] ; the symmetric-slot realization of higher spin is described by Nachtergaele [13 ] . The pin argument uses the stable-polynomial approach to negative dependence developed by Borcea, Brändén, and Liggett [3 ] . Diffusive smoothing uses the energy method of Nash [14 ] . The companion [15 ] provides the particular stable exchange symbols, presence inequality, local lattice inequality, and routed unit flows used below. We prove here the finite-range pin bootstrap, the quantitative long-return bound, and the comparison that recovers the full dispersion.
How returns determine the coefficient
Represent each spin by ℓ = 2 S \ell=2S ℓ = 2 S 1 / 2 1/2 1/2 β \beta β pin law . If E E E i i i
d E ( i ) = r ( E , { i } ) 1 + r ( E , { i } ) . d_E(i)=\frac{r(E,\{i\})}{1+r(E,\{i\})}. d E ( i ) = 1 + r ( E , { i }) r ( E , { i }) . where d E ( i ) d_E(i) d E ( i ) i i i r ( E , { i } ) r(E,\{i\}) r ( E , { i }) i i i i i i E E E r r r
There are two geometric tasks. For rough estimates, choose three linearly independent interaction vectors. They generate a sublattice of finite index; on each of its cosets, the chosen bonds form a copy of the ordinary cubic lattice. Section 3 transfers local Nash and flow estimates to these copies and compares their energies with the full interaction energy. Section 4 then proves a sparse-pin estimate by downward induction on the pin set. This induction supplies enough regular time for diffusion without assuming a finite-range ordering theorem.
Sharp asymptotics require all interaction bonds. The ideal walk has site jump rates S J ( z ) S J(z) S J ( z ) p t S , J p_t^{S,J} p t S , J
p t S , J ( x , x ) ∼ κ t − 3 / 2 , κ = ( 4 π ) − 3 / 2 ( det D S , J ) − 1 / 2 . p_t^{S,J}(x,x) \sim\kappa t^{-3/2}, \qquad\kappa= (4\pi)^{-3/2}(\det D_{S,J})^{-1/2}. p t S , J ( x , x ) ∼ κ t − 3/2 , κ = ( 4 π ) − 3/2 ( det D S , J ) − 1/2 . After a full traversal the slot index is uniform, so the ideal n n n ℓ − 1 p n β S , J ( x , x ) \ell^{-1}p_{n\beta}^{S,J}(x,x) ℓ − 1 p n β S , J ( x , x ) ℓ \ell ℓ κ ∑ n ≤ K n − 3 / 2 \kappa\sum_{n\le K} n^{-3/2} κ ∑ n ≤ K n − 3/2
The main analytic difficulty is controlling arbitrarily late returns in the same repeated environment. Section 5 truncates a quadratic hitting-probability identity after K K K K β K\beta K β C ( K β ) − 1 / 2 C(K\beta)^{-1/2} C ( K β ) − 1/2 C K − 1 / 2 β − 3 / 2 + o K ( β − 3 / 2 ) C K^{-1/2}\beta^{-3/2}+o_K(\beta^{-3/2}) C K − 1/2 β − 3/2 + o K ( β − 3/2 ) C C C K K K K = 1 K=1 K = 1 O ( β − 3 / 2 ) O(\beta^{-3/2}) O ( β − 3/2 )
This improved density permits the comparison in Section 6. The ideal path rarely encounters an exposed strand over a fixed number of periods. Splitting a return at its midpoint combines that small encounter probability with diffusive smoothing, giving an error o K ( β − 3 / 2 ) o_K(\beta^{-3/2}) o K ( β − 3/2 )
Finally, Section 7 expresses a positive field as a mixture of extra pins. The probability of selecting any given slot is at most 1 − e − β h 1-e^{-\beta h} 1 − e − β h h ↓ 0 h\downarrow0 h ↓ 0 β , K \beta,K β , K
Exchange pictures and pinned returns
We express the magnetization through colored exchange cycles, then turn pinning probabilities into one-particle return probabilities. Two changes of law are involved. Pinning removes the color factor from cycles that meet the pins. Exposing the cycles that avoid the pins leaves a conditional exchange process on the remaining slots. The last part of this section identifies its first return moment with an unconditioned Markov walk in the exposed environment. These constructions are finite-dimensional; the spatial estimates begin in Section 3.
Spin slots and the exchange law
Fix β > 0 \beta>0 β > 0 Λ = Λ N \Lambda=\Lambda_N Λ = Λ N ℓ = 2 S \ell=2S ℓ = 2 S
b 0 = max { ∣ z ∣ ∞ : J ( z ) > 0 } , ∂ Λ = { x ∈ Λ : dist ∞ ( x , Z 3 ∖ Λ ) ≤ b 0 } , Λ ∘ = Λ ∖ ∂ Λ . b_0=\max\{|z|_\infty:J(z)>0\},\qquad\partial\Lambda=\{x\in\Lambda:\operatorname{dist}_\infty(x,\mathbb{Z}^3\setminus\Lambda)\le b_0\},\qquad\Lambda^\circ=\Lambda\setminus\partial\Lambda. b 0 = max { ∣ z ∣ ∞ : J ( z ) > 0 } , ∂ Λ = { x ∈ Λ : dist ∞ ( x , Z 3 ∖ Λ ) ≤ b 0 } , Λ ∘ = Λ ∖ ∂ Λ. No interaction bond joins Λ ∘ \Lambda^{\circ} Λ ∘ Λ \Lambda Λ V Λ = Λ × { 1 , … , ℓ } V_{\Lambda}=\Lambda\times\{1,\ldots,\ell\} V Λ = Λ × { 1 , … , ℓ } { x , y } ⊂ Λ \{x,y\}\subset\Lambda { x , y } ⊂ Λ J ( y − x ) > 0 J(y-x)>0 J ( y − x ) > 0 ℓ 2 \ell^2 ℓ 2 E Λ \mathcal{E}_{\Lambda} E Λ
J e = J ( y − x ) , c e = J e 2 ( e above { x , y } ) , w Λ = ∑ e ∈ E Λ J e . J_e=J(y-x),\qquad c_e=\frac{J_e}{2}\quad(e\text{ above }\{x,y\}),\qquad w_{\Lambda}=\sum_{e\in\mathcal{E}_{\Lambda}}J_e. J e = J ( y − x ) , c e = 2 J e ( e above { x , y }) , w Λ = e ∈ E Λ ∑ J e . Under the base law P b \mathbb{P}_{\mathrm b} P b e e e c e c_e c e [ 0 , β ) [0,\beta) [ 0 , β ) β \beta β 0 0 0 seam completions . A picture ω \omega ω V Λ V_{\Lambda} V Λ
Time belongs to R / β Z \mathbb{R}/\beta\mathbb{Z} R / β Z a C a_C a C C C C ∑ C a C = ℓ ∣ Λ ∣ \sum_C a_C=\ell|\Lambda| ∑ C a C = ℓ ∣Λ∣ σ C ∈ { − 1 , 1 } \sigma_C\in\{-1,1\} σ C ∈ { − 1 , 1 }
M ( σ ) = 1 2 ∑ C a C σ C . M(\sigma)=\frac{1}{2}\sum_C a_C\sigma_C. M ( σ ) = 2 1 C ∑ a C σ C . Proposition 2.1 (Loop representation). For every finite cube, every β > 0 \beta>0 β > 0 h h h
Z Λ ( h ) : = Tr e − β ( H S , N J − h M S , N ) = e β w Λ / 4 E b ∑ σ e β h M ( σ ) . (5) Z_{\Lambda}(h):=\operatorname{Tr}e^{-\beta(H^J_{S,N}-hM_{S,N})}=e^{\beta w_{\Lambda}/4}\mathbb{E}_{\mathrm b}\sum_{\sigma}e^{\beta hM(\sigma)}. \tag*{(5)} Z Λ ( h ) := Tr e − β ( H S , N J − h M S , N ) = e β w Λ /4 E b σ ∑ e β h M ( σ ) . (5) At zero field the Gibbs spectral law of total magnetization is obtained by weighting each picture by 2 # { cycles } 2^{\#\{\text{cycles}\}} 2 # { cycles } M ( σ ) M(\sigma) M ( σ ) Φ \Phi Φ M S , N M_{S,N} M S , N
Tr ( e − β H S , N J Φ ( M S , N ) ) Z Λ ( 0 ) = E b ∑ σ Φ ( M ( σ ) ) E b 2 # { cycles } . (6) \frac{\operatorname{Tr}(e^{-\beta H^J_{S,N}}\Phi(M_{S,N}))}{Z_{\Lambda}(0)}
=\frac{\mathbb{E}_{\mathrm b}\sum_{\sigma}\Phi(M(\sigma))}{\mathbb{E}_{\mathrm b}2^{\#\{\text{cycles}\}}}. \tag*{(6)} Z Λ ( 0 ) Tr ( e − β H S , N J Φ ( M S , N )) = E b 2 # { cycles } E b ∑ σ Φ ( M ( σ )) . (6) Proof. This is the exchange representation of Tóth [17 ] , with the higher-spin symmetric-slot construction of Nachtergaele [13 ] , Section 2.2; see also [15 ] , Lemma 2.1. At one site let ψ k \psi_k ψ k 1 2 \frac{1}{2} 2 1 k k k 0 ≤ k ≤ ℓ 0\leq k\leq\ell 0 ≤ k ≤ ℓ z z z ψ k \psi_k ψ k k − ℓ / 2 k-\ell/2 k − ℓ /2 ( ℓ − k ) ( k + 1 ) ψ k + 1 \sqrt{(\ell-k)(k+1)}\psi_{k+1} ( ℓ − k ) ( k + 1 ) ψ k + 1 S S S k = S + m k=S+m k = S + m
S Λ = ∏ x ∈ Λ S x , S x = 1 ℓ ! ∑ κ ∈ S ℓ U κ , x , S_{\Lambda}=\prod_{x\in\Lambda}S_x,\qquad S_x=\frac{1}{\ell!}\sum_{\kappa\in\mathfrak{S}_{\ell}}U_{\kappa,x}, S Λ = x ∈ Λ ∏ S x , S x = ℓ ! 1 κ ∈ S ℓ ∑ U κ , x , where U U U 1 2 \frac{1}{2} 2 1 s i ⋅ s j = 1 2 U ( i j ) − 1 4 I s_i\cdot s_j=\frac{1}{2}U_{(ij)}-\frac{1}{4}I s i ⋅ s j = 2 1 U ( ij ) − 4 1 I
H ^ = − 1 2 ∑ e J e U e + 1 4 w Λ I , M ^ = ∑ i ∈ V Λ s i z . \widehat{H}=-\frac{1}{2}\sum_e J_eU_e+\frac{1}{4}w_{\Lambda}I,\qquad\widehat{M}=\sum_{i\in V_{\Lambda}}s_i^z. H = − 2 1 e ∑ J e U e + 4 1 w Λ I , M = i ∈ V Λ ∑ s i z . The full sum over all slot pairs above each bond commutes with every within-site permutation, and hence with S Λ S_{\Lambda} S Λ φ \varphi φ
E b U φ = exp { β ∑ e c e ( U e − I ) } = e − β w Λ / 4 e − β H ^ . \mathbb{E}_{\mathrm b}U_{\varphi}=\exp\left\{\beta\sum_e c_e(U_e-I)\right\}=e^{-\beta w_{\Lambda}/4}e^{-\beta\widehat{H}}. E b U φ = exp { β e ∑ c e ( U e − I ) } = e − β w Λ /4 e − β H . For example, the first equality follows from the matrix evolution with generator ∑ e c e ( U e − I ) \sum_e c_e(\mathcal{U}_e-I) ∑ e c e ( U e − I ) S Λ S_\Lambda S Λ e β M ^ e^{\beta\widehat{M}} e β M M ^ \widehat{M} M
Pins, boundary approximation, and mark counts
A pin is a specified slot-time point. Define the continuous boundary pin set
D 0 = ∂ Λ × { 1 , … , ℓ } × ( R / β Z ) . D_0=\partial\Lambda\times\{1,\ldots,\ell\}\times(\mathbb{R}/\beta\mathbb{Z}). D 0 = ∂ Λ × { 1 , … , ℓ } × ( R / β Z ) . An allowed pin set is D = D 0 ∪ F D=D_0\cup F D = D 0 ∪ F F F F
n D ( ω ) = # { C : C ∩ D = ∅ } , B D ( ω ) = ⋃ C : C ∩ D = ∅ C , Z D = E b 2 n D , n_D(\omega)=\#\{C:C\cap D=\varnothing\},\qquad\mathcal{B}_D(\omega)=\bigcup_{C:C\cap D=\varnothing}C,\qquad Z_D=\mathbb{E}_{\mathrm{b}}2^{n_D}, n D ( ω ) = # { C : C ∩ D = ∅ } , B D ( ω ) = C : C ∩ D = ∅ ⋃ C , Z D = E b 2 n D , and define
P D ( d ω ) = Z D − 1 2 n D ( ω ) P b ( d ω ) . (7) \mathbb{P}_D(d\omega)=Z_D^{-1}2^{n_D(\omega)}\mathbb{P}_{\mathrm{b}}(d\omega). \tag*{(7)} P D ( d ω ) = Z D − 1 2 n D ( ω ) P b ( d ω ) . (7) The normalizer obeys 1 ≤ Z D ≤ 2 ∣ V Λ ∣ 1\le Z_D\le2^{|V_\Lambda|} 1 ≤ Z D ≤ 2 ∣ V Λ ∣ D D D d D ( i ) d_D(i) d D ( i ) i i i
2 d D ( i ) = P D ( i ∈ B D ) . (8) 2d_D(i)=\mathbb{P}_D(i\in\mathcal{B}_D). \tag*{(8)} 2 d D ( i ) = P D ( i ∈ B D ) . (8) Lemma 2.2 (Dense boundary cuts). Let F F F T n T_n T n
D n = F ∪ ( ∂ Λ × { 1 , … , ℓ } × T n ) , D = F ∪ D 0 . D_n=F\cup(\partial\Lambda\times\{1,\ldots,\ell\}\times T_n),\qquad D=F\cup D_0. D n = F ∪ ( ∂ Λ × { 1 , … , ℓ } × T n ) , D = F ∪ D 0 . Almost surely, n D n = n D n_{D_n}=n_D n D n = n D B D n = B D \mathcal{B}_{D_n}=\mathcal{B}_D B D n = B D n n n P D n \mathbb{P}_{D_n} P D n P D \mathbb{P}_D P D X n X_n X n X X X E D n X n → E D X \mathbb{E}_{D_n}X_n\to\mathbb{E}_D X E D n X n → E D X
Proof. There are finitely many marks per period, their times are distinct, and seam completions preserve sites. Every visit to the boundary therefore contains a time interval of positive length and is detected by a cut from the dense union. There are finitely many cycles, so one finite cut set eventually detects every boundary-touching cycle. The asserted eventual equalities follow. Since 1 ≤ 2 n D n ≤ 2 ∣ V Λ ∣ 1\le2^{n_{D_n}}\le2^{|V_\Lambda|} 1 ≤ 2 n D n ≤ 2 ∣ V Λ ∣ L 1 ( P b ) L^1(\mathbb{P}_{\mathrm{b}}) L 1 ( P b )
Under a pin law the marks need not be independent. The following uniform estimate on lists of distinct marks will suffice for all path counts. Let ω m \omega_m ω m X Λ = E Λ × [ 0 , β ) X_\Lambda=E_\Lambda\times[0,\beta) X Λ = E Λ × [ 0 , β )
λ b ( { e } × d t ) = c e d t . \lambda_{\mathrm{b}}(\{e\}\times dt)=c_e\,dt. λ b ({ e } × d t ) = c e d t . Lemma 2.3 (Factorial domination). For an allowed or finite pin set D D D j ≥ 1 j\ge1 j ≥ 1 F F F X Λ j X_\Lambda^j X Λ j
E D ∑ ( u 1 , … , u j ) ∈ ( ω m ) ≠ j F ( u 1 , … , u j ) ≤ 2 j ∫ F d λ b ⊗ j . (9) \mathbb{E}_D\sum_{(u_1,\ldots,u_j)\in(\omega_m)^j_{\ne}}F(u_1,\ldots,u_j)\le2^j\int F\,d\lambda_{\mathrm{b}}^{\otimes j}. \tag*{(9)} E D ( u 1 , … , u j ) ∈ ( ω m ) = j ∑ F ( u 1 , … , u j ) ≤ 2 j ∫ F d λ b ⊗ j . (9) Here the sum is over ordered tuples of distinct marks. The estimate also holds with any prescribed time order or time restrictions.
Proof. Inserting a transposition cuts and rejoins two strands. It joins two cycles or splits one cycle into two, leaving the other cycles unchanged, and preserves the union of the affected strands and their pin incidences. Joining cycles cannot increase n D n_D n D D D D D D D D D D n D n_D n D
The Poisson insertion identity, with the seam fixed, is
E b [ 2 n D ( ω ) ∑ ( u 1 , … , u j ) ∈ ( ω m ) j ≠ F ( u 1 , … , u j ) ] = ∫ F ( u 1 , … , u j ) E b 2 n D ( ω + ∑ a = 1 j δ u a ) ∏ a = 1 j λ b ( d u a ) . \mathbb{E}_b\left[2^{n_D(\omega)}\sum_{(u_1,\ldots,u_j)\in(\omega_m)^j}^{\ne}F(u_1,\ldots,u_j)\right]
=
\int F(u_1,\ldots,u_j)\mathbb{E}_b 2^{n_D\left(\omega+\sum_{a=1}^{j}\delta_{u_a}\right)}\prod_{a=1}^{j}\lambda_b(du_a). E b 2 n D ( ω ) ( u 1 , … , u j ) ∈ ( ω m ) j ∑ = F ( u 1 , … , u j ) = ∫ F ( u 1 , … , u j ) E b 2 n D ( ω + ∑ a = 1 j δ u a ) a = 1 ∏ j λ b ( d u a ) . Indeed, in the Poisson series the k k k k ! / ( k − j ) ! k!/(k-j)! k ! / ( k − j )! 1 / k ! 1/k! 1/ k ! Z D Z_D Z D
We shall freely use the weaker constant 4 j 4^j 4 j 4 λ b 4\lambda_b 4 λ b λ b \lambda_b λ b P D \mathbb{P}_D P D
Single and simultaneous pin tests
Let E E E T T T m T = ∣ T ∣ m_T=|T| m T = ∣ T ∣ θ \theta θ E E E P = E ∪ T P=E\cup T P = E ∪ T R i ( E , T ) \mathcal{R}_i(E,T) R i ( E , T ) i ∈ T i\in T i ∈ T T T T E E E
r ( E , T ) = 1 m T ∑ i ∈ T P P ( R i ( E , T ) ) . (10) r(E,T)=\frac{1}{m_T}\sum_{i\in T}\mathbb{P}_P(\mathcal{R}_i(E,T)). \tag*{(10)} r ( E , T ) = m T 1 i ∈ T ∑ P P ( R i ( E , T )) . (10) Pinning T T T T T T
Proposition 2.4 (Pin identities). For E , T E,T E , T
d E ( i ) = r ( E , { i } ) 1 + r ( E , { i } ) ( i ∉ E ) , P E ( T ⊆ B E ) ≤ ( 2 r ( E , T ) ) m T . (11) d_E(i)=\frac{r(E,\{i\})}{1+r(E,\{i\})}\quad(i\notin E),\qquad\mathbb{P}_E(T\subseteq\mathcal{B}_E)\leq\bigl(2r(E,T)\bigr)^{m_T}. \tag*{(11)} d E ( i ) = 1 + r ( E , { i }) r ( E , { i }) ( i ∈ / E ) , P E ( T ⊆ B E ) ≤ ( 2 r ( E , T ) ) m T . (11) Proof. For T = { i } T=\{i\} T = { i } r = r ( E , { i } ) r=r(E,\{i\}) r = r ( E , { i }) A = { i ∈ B E } A=\{i\in\mathcal{B}_E\} A = { i ∈ B E } i i i E E E A A A r ( E , { i } ) = P P ( A ) r(E,\{i\})=\mathbb{P}_P(A) r ( E , { i }) = P P ( A ) 2 n E = 2 n P ( 1 + 1 A ) 2^{n_E}=2^{n_P}(1+1_A) 2 n E = 2 n P ( 1 + 1 A ) Z E = Z P ( 1 + r ) Z_E=Z_P(1+r) Z E = Z P ( 1 + r ) P E ( A ) = 2 r / ( 1 + r ) \mathbb{P}_E(A)=2r/(1+r) P E ( A ) = 2 r / ( 1 + r ) (8) proves the singleton identity. This calculation applies directly to finite and allowed pin sets.
For the simultaneous bound, we use three precise finite-set facts from [15 ] , Lemmas 3.2, 3.4, and 3.6. A nonzero polynomial is stable if it has no zero when every variable has positive imaginary part. The facts are as follows.
(i) Variable identification preserves stability. Partial derivatives, real specialization, and coefficientwise limits of uniformly bounded degree preserve stability up to the zero polynomial. In particular, after identifying two variables as t t t t t t
(ii) If a permutation π \pi π V V V
E ∏ j ∈ V ( x j + y π ( j ) ) \mathbb{E}\prod_{j\in V}(x_j+y_{\pi(j)}) E j ∈ V ∏ ( x j + y π ( j ) ) is stable. This also holds for independent products of finite-time Poisson exchanges with arbitrary nonnegative rates and uniform permutations on specified subsets.
(iii) If a random subset A A A E ∏ j ∈ A y j \mathbb{E}\prod_{j\in A} y_j E ∏ j ∈ A y j B B B
Pr ( B ⊆ A ) ≤ ∏ j ∈ B Pr ( j ∈ A ) . \Pr(B\subseteq A)\leq\prod_{j\in B}\Pr(j\in A). Pr ( B ⊆ A ) ≤ j ∈ B ∏ Pr ( j ∈ A ) . These facts belong to the stable-polynomial method for negative dependence developed in [3 ] . They concern arbitrary finite sets, with no spatial hypothesis. To check the continuous-time scope in (ii), subdivide a finite interval into small slices and replace each edge clock in each slice by its independent indicator of at least one mark. Apply these Bernoulli transpositions in a fixed order. The products agree with the Poisson products whenever no slice has two marks in total; the probability of disagreement tends to zero. A uniform permutation is likewise a distributional limit of complete-graph Poisson exchanges: the unit-time transition matrix on its finite permutation group has all entries positive and preserves the uniform law, so it contracts total variation distance to that law. Thus the rates c e = J e / 2 c_e=J_e/2 c e = J e /2
For this bound, first suppose that E E E P P P V V V π \pi π x x x y y y
F V = E b ∏ j ∈ V ( x j + y π ( j ) ) . F_V=\mathbb{E}_{b}\prod_{j\in V}(x_j+y_{\pi(j)}). F V = E b j ∈ V ∏ ( x j + y π ( j ) ) . Delete an unpinned point q ∈ V ∖ P q\in V\setminus P q ∈ V ∖ P x q = y q = t x_q=y_q=t x q = y q = t t t t i i i j j j q q q ( x i + t ) ( t + y j ) (x_i+t)(t+y_j) ( x i + t ) ( t + y j ) x i + y j x_i+y_j x i + y j i → q → j i\to q\to j i → q → j i → j i\to j i → j 2 t 2t 2 t
F P = E b [ 2 n P ∏ j ∈ P ( x j + y σ ( j ) ) ] (12) F_P=\mathbb{E}_{b}\left[2^{n_P}\prod_{j\in P}(x_j+y_{\sigma(j)})\right] \tag*{(12)} F P = E b 2 n P j ∈ P ∏ ( x j + y σ ( j ) ) (12) where σ \sigma σ P P P
Differentiate (2.8) in every x j x_j x j j ∈ P ∖ T j\in P\setminus T j ∈ P ∖ T x x x Z P Z_P Z P
G T ( y ) = E P ∏ i ∈ T y σ ( i ) . (13) G_T(\mathbf{y})=\mathbb{E}_{P}\prod_{i\in T}y_{\sigma(i)}. \tag*{(13)} G T ( y ) = E P i ∈ T ∏ y σ ( i ) . (13) It is nonzero because G T ( 1 ) = 1 G_T(1)=1 G T ( 1 ) = 1 σ ( T ) \sigma(T) σ ( T ) T T T
m T r ( E , T ) = E P ∣ T ∩ σ ( T ) ∣ = ∑ j ∈ T P ( j ∈ σ ( T ) ) . m_T r(E,T)=\mathbb{E}_{P}|T\cap\sigma(T)|=\sum_{j\in T}\mathbb{P}(j\in\sigma(T)). m T r ( E , T ) = E P ∣ T ∩ σ ( T ) ∣ = j ∈ T ∑ P ( j ∈ σ ( T )) . Fact (iii) and the arithmetic–geometric mean inequality now imply
P P ( T ⊆ σ ( T ) ) ≤ r ( E , T ) m T . \mathbb{P}_{P}(T\subseteq\sigma(T))\leq r(E,T)^{m_T}. P P ( T ⊆ σ ( T )) ≤ r ( E , T ) m T . The event T ⊆ σ ( T ) T\subseteq\sigma(T) T ⊆ σ ( T ) T ⊆ B E T\subseteq B_E T ⊆ B E ∣ σ ( T ) ∣ = ∣ T ∣ |\sigma(T)|=|T| ∣ σ ( T ) ∣ = ∣ T ∣ σ ( T ) = T \sigma(T)=T σ ( T ) = T σ \sigma σ T T T T T T
Deleting m T m_T m T m T m_T m T
2 n P ≤ 2 n E ≤ 2 m T 2 n P , Z E ≥ Z P > 0. 2^{n^P}\leq2^{n^E}\leq2^{m_T}2^{n^P},\qquad Z_E\geq Z_P>0. 2 n P ≤ 2 n E ≤ 2 m T 2 n P , Z E ≥ Z P > 0. For A = { T ⊆ B E } A=\{T\subseteq B_E\} A = { T ⊆ B E }
P E ( A ) ≤ 2 m T Z P Z E P P ( A ) ≤ ( 2 r ( E , T ) ) m T . \mathbb{P}_E(A)\leq2^{m_T}\frac{Z_P}{Z_E}\mathbb{P}_P(A)\leq(2r(E,T))^{m_T}. P E ( A ) ≤ 2 m T Z E Z P P P ( A ) ≤ ( 2 r ( E , T ) ) m T . For E = F ∪ D 0 E=F\cup D_0 E = F ∪ D 0 D 0 D_0 D 0 E n E_n E n P n = E n ∪ T P_n=E_n\cup T P n = E n ∪ T i ∈ T i\in T i ∈ T T T T i i i 1 R i ( E n , T ) → 1 R i ( E , T ) \mathbf{1}_{R_i}(E_n,T)\to\mathbf{1}_{R_i}(E,T) 1 R i ( E n , T ) → 1 R i ( E , T )
Exposing the cycles that miss the pins
The simultaneous pin inequality reduces the avoidance event to an average return under P P \mathbb{P}_P P P E E E T T T P P P P P \mathbb{P}_P P P P P P a a a H \mathcal{H} H t t t H t = H t ( a ) H_t=H_t(a) H t = H t ( a )
For fixed a a a Q a \mathbb{Q}_a Q a c E c_E c E forced transport is a fixed bijection between the hole layers before and after the exposed jump. Exchanges of two exposed labels do not affect holes. At the seam, independently at each site, complete the revealed mapping by a uniform bijection between its remaining source and target slots. Let G P G_P G P P P P
g P ( a ) = Q a ( G P ) . g_P(a)=\mathbb{Q}_a(G_P). g P ( a ) = Q a ( G P ) . Proposition 2.5 (Exposure disintegration). Almost surely under P P \mathbb{P}_P P P g P ( a ) > 0 g_P(a)>0 g P ( a ) > 0 H \mathcal{H} H Q a ( ⋅ ∣ G P ) \mathbb{Q}_a(\,\cdot\mid G_P) Q a ( ⋅ ∣ G P )
More precisely, for a deterministic initial label set A ⊆ V Λ A \subseteq V_\Lambda A ⊆ V Λ μ A \mu_A μ A F A F_A F A P P P A A A F A F_A F A c A ( a ) c_A(a) c A ( a ) Φ \Phi Φ ω h \omega_h ω h
E P Φ ( a , ω h ) = 1 Z P ∑ A ⊆ V Λ ∫ F A 2 c A ( a ) E Q a [ 1 G P Φ ( a , ω h ) ] μ A ( d a ) . (14) \mathbb{E}_{P}\Phi(a,\omega_h)=\frac{1}{Z_P}\sum_{A\subseteq V_\Lambda}\int_{F_A}2^{c_A(a)}\mathbb{E}_{Q_a}\left[1_{G_P}\Phi(a,\omega_h)\right]\mu_A(da). \tag*{(14)} E P Φ ( a , ω h ) = Z P 1 A ⊆ V Λ ∑ ∫ F A 2 c A ( a ) E Q a [ 1 G P Φ ( a , ω h ) ] μ A ( d a ) . (14) In particular, the exposure marginal on this branch is
Z P − 1 1 F A 2 c A ( a ) g P ( a ) μ A ( d a ) . (15) Z_P^{-1}1_{F_A}2^{c_A(a)}g_P(a)\,\mu_A(da). \tag*{(15)} Z P − 1 1 F A 2 c A ( a ) g P ( a ) μ A ( d a ) . (15) The same assertions hold for finite deterministic P P P
Proof. We give the rate-dependent construction underlying [15 ] . Fix A A A e e e c e c_e c e A A A
This produces the base exchange process. Indeed, if c t o t = ∑ e c e > 0 c_{\mathrm{tot}}=\sum_e c_e>0 c tot = ∑ e c e > 0 2 c t o t 2c_{\mathrm{tot}} 2 c tot c e / ( 2 c t o t ) c_e/(2c_{\mathrm{tot}}) c e / ( 2 c tot ) e e e 1 / 2 1/2 1/2 c e c_e c e c t o t = 0 c_{\mathrm{tot}}=0 c tot = 0
The A A A Q a Q_a Q a F A F_A F A
On F A F_A F A A A A P P P G P G_P G P
2 n P = ∑ A ⊆ V Λ 1 F A 2 c A 1 G P . 2^{n_P}=\sum_{A\subseteq V_\Lambda}1_{F_A}2^{c_A}1_{G_P}. 2 n P = A ⊆ V Λ ∑ 1 F A 2 c A 1 G P . where a summand is zero off F A F_A F A Z P Z_P Z P g P = 0 g_P=0 g P = 0 g P g_P g P
The factor g P g_P g P P P \mathbb{P}_P P P Q a Q_a Q a
Removing the cycle condition from a first moment
We record the finite-dimensional argument from [15 ] , including its proof to specify precisely why fresh transitions appear. Matrices use the column-forward convention: a permutation matrix obeys Π e i = e π ( i ) \Pi e_i=e_{\pi(i)} Π e i = e π ( i ) M M M ⋀ k M \bigwedge^k M ⋀ k M k k k k × k k\times k k × k M M M ⋀ 0 M = 1 \bigwedge^0 M=1 ⋀ 0 M = 1
Lemma 2.6 (Exterior moments). Suppose a random permutation matrix Π \Pi Π on a finite set is a distributional limit of products of independent Bernoulli transpositions and deterministic permutations. If K = E Π K = \mathbb{E}\Pi K = E Π then
E ⋀ k Π = ⋀ k K ( 0 ≤ k ≤ ∣ V ∣ ) . (16) \mathbb{E}\bigwedge^{k}\Pi= \bigwedge^{k}K \qquad(0 \le k \le|V|). \tag*{(16)} E ⋀ k Π = ⋀ k K ( 0 ≤ k ≤ ∣ V ∣ ) . (16) For every fixed exposure and finite set of cuts, the layered permutation obtained under Q a \mathbb{Q}_{a} Q a by following holes to the next cyclic cut has this property.
Proof. For a transposition matrix T i j T_{ij} T ij
T i j − I = − ( e i − e j ) ( e i − e j ) ∗ . T_{ij}-I = -(e_i-e_j)(e_i-e_j)^{*}. T ij − I = − ( e i − e j ) ( e i − e j ) ∗ . Every minor of I + u ( T i j − I ) I+u(T_{ij}-I) I + u ( T ij − I ) u u u
⋀ k ( ( 1 − u ) I + u T i j ) = ( 1 − u ) ⋀ k I + u ⋀ k T i j , \bigwedge^{k}\bigl((1-u)I+uT_{ij}\bigr)=(1-u)\bigwedge^{k}I+u\bigwedge^{k}T_{ij}, ⋀ k ( ( 1 − u ) I + u T ij ) = ( 1 − u ) ⋀ k I + u ⋀ k T ij , which proves the identity for one Bernoulli transposition. Multiplicativity of exterior powers and independence preserve it under products. Deterministic permutations satisfy it directly, and continuity of the minors on the finite matrix space passes it to distributional limits.
For the hole picture, subdivide each interval between prescribed changes and use the Bernoulli approximation already described. Forced transports are deterministic bijections. The residual seam completions are uniform bijections, each expressible as a fixed bijection composed with a uniform permutation; the latter has the complete-graph exchange approximation described above. Identify every cut layer with a fixed set of ∣ H t ∣ |H_t| ∣ H t ∣ G P G_P G P
Proposition 2.7 (Conditional next-pin kernel). Let V = I ⊔ B V=I\sqcup B V = I ⊔ B I ≠ ∅ I\ne\varnothing I = ∅ Π \Pi Π K = E Π K=\mathbb{E}\Pi K = E Π G I G_I G I I I I G I G_I G I Σ \Sigma Σ I I I I I I
Pr ( G I ) = det ( I − K B B ) . \Pr(G_I)=\det(I-K_{BB}). Pr ( G I ) = det ( I − K B B ) . If this probability is positive, then K B B K_{BB} K B B
E [ Σ ∣ G I ] = K I I + K I B ( I − K B B ) − 1 K B I . (17) \mathbb{E}[\Sigma\mid G_I]=K_{II}+K_{IB}(I-K_{BB})^{-1}K_{BI}. \tag*{(17)} E [ Σ ∣ G I ] = K I I + K I B ( I − K B B ) − 1 K B I . (17) The right-hand side is the first positive I I I hitting kernel of the Markov chain with independent transitions according to K K K If B B B is empty, the determinant is one and the correction is zero.
Proof. The principal-minor expansion and (2.12) give, for every fixed matrix D D D
E det ( I − D Π ) = ∑ k ( − 1 ) k Tr [ ( ⋀ k D ) E ⋀ k Π ] = det ( I − D K ) . (18) \mathbb{E}\det(I-D\Pi)=\sum_k(-1)^k\operatorname{Tr}\left[\left(\bigwedge^k D\right)\mathbb{E}\bigwedge^k\Pi\right]=\det(I-DK). \tag*{(18)} E det ( I − D Π ) = k ∑ ( − 1 ) k Tr [ ( ⋀ k D ) E ⋀ k Π ] = det ( I − D K ) . (18) Choose D = diag ( W , I B ) D=\operatorname{diag}(W,I_B) D = diag ( W , I B ) I × I I\times I I × I W W W B B B D Π D\Pi D Π G I G_I G I Π B B \Pi_{BB} Π B B
det ( I − D Π ) = det ( I − W [ Π I I + Π I B ( I − Π B B ) − 1 Π B I ] ) = det ( I − W Σ ) . \det(I-D\Pi)=\det\left(I-W\left[\Pi_{II}+\Pi_{IB}(I-\Pi_{BB})^{-1}\Pi_{BI}\right]\right)=\det(I-W\Sigma). det ( I − D Π ) = det ( I − W [ Π I I + Π I B ( I − Π B B ) − 1 Π B I ] ) = det ( I − W Σ ) . Consequently
E [ 1 G I det ( I − W Σ ) ] = det ( I − W K I I − W K I B − K B I I − K B B ) (19) \mathbb{E}[1_{G_I}\det(I-W\Sigma)] = \det\begin{pmatrix} I-WK_{II} & -WK_{IB} \\ -K_{BI} & I-K_{BB} \end{pmatrix} \tag*{(19)} E [ 1 G I det ( I − W Σ )] = det ( I − W K I I − K B I − W K I B I − K B B ) (19) Setting W = 0 W=0 W = 0 I − K B B I-K_{BB} I − K B B
E [ det ( I − W Σ ) ∣ G I ] = det ( I − W H ) , H = K I I + K I B ( I − K B B ) − 1 K B I . \mathbb{E}[\det(I-W\Sigma)\mid G_I] = \det(I-WH), \qquad H=K_{II}+K_{IB}(I-K_{BB})^{-1}K_{BI}. E [ det ( I − W Σ ) ∣ G I ] = det ( I − W H ) , H = K I I + K I B ( I − K B B ) − 1 K B I . Replace W W W t W tW t W det ( I − t W A ) = 1 − t Tr ( W A ) + O ( t 2 ) \det(I-tWA)=1-t\operatorname{Tr}(WA)+O(t^2) det ( I − t W A ) = 1 − t Tr ( W A ) + O ( t 2 ) Tr ( W E [ Σ ∣ G I ] ) = Tr ( W H ) \operatorname{Tr}(W\mathbb{E}[\Sigma\mid G_I])=\operatorname{Tr}(WH) Tr ( W E [ Σ ∣ G I ]) = Tr ( W H ) W W W
Finally, K K K K B B K_{BB} K B B B B B I I I K B B K_{BB} K B B a a a ε > 0 \varepsilon>0 ε > 0 B B B a a a ε \varepsilon ε ∥ K B B a n ∥ 1 → 1 ≤ ( 1 − ε ) n \lVert K_{BB}^{an}\rVert_{1\to1}\le(1-\varepsilon)^n ∥ K B B an ∥ 1 → 1 ≤ ( 1 − ε ) n
( I − K B B ) − 1 = ∑ n ≥ 0 K B B n . (I-K_{BB})^{-1}=\sum_{n\ge0}K_{BB}^n. ( I − K B B ) − 1 = n ≥ 0 ∑ K B B n . The first term in H H H I I I K I B K B B n K B I K_{IB}K_{BB}^nK_{BI} K I B K B B n K B I n + 1 n+1 n + 1 B B B
The fresh walk
Fix an exposure with g P ( a ) > 0 g_P(a)>0 g P ( a ) > 0 fresh walk follows the one-particle transitions of Q a Q_a Q a c e c_e c e i ∈ T i\in T i ∈ T τ T + \tau_T^+ τ T + T T T τ E \tau_E τ E E E E
q i ( a ) = P a , i w a l k ( τ T + < τ E ) . (20) q_i(a)=\mathbb{P}^{\mathrm{walk}}_{a,i}(\tau_T^+<\tau_E). \tag*{(20)} q i ( a ) = P a , i walk ( τ T + < τ E ) . (20) Proposition 2.8 (Fresh-walk identity). For almost every exposure under P P \mathbb{P}_P P P i ∈ T i\in T i ∈ T
P P ( R i ( E , T ) ∣ H ) = q i ( a ) . \mathbb{P}_P(\mathcal{R}_i(E,T)\mid\mathcal{H})=q_i(a). P P ( R i ( E , T ) ∣ H ) = q i ( a ) . In particular,
m T r ( E , T ) = E P ∑ i ∈ T q i ( a ) . (21) m_T r(E,T)=\mathbb{E}_P\sum_{i\in T}q_i(a). \tag*{(21)} m T r ( E , T ) = E P i ∈ T ∑ q i ( a ) . (21) Proof. First use finitely many pins and take cuts at all pin times and zero. The unconditioned layered hole permutation under Q a Q_a Q a
For an allowed E = F ∪ D 0 E=F\cup D_0 E = F ∪ D 0 g P > 0 g_P>0 g P > 0 E n = F ∪ D 0 , n E_n=F\cup D_{0,n} E n = F ∪ D 0 , n P n = E n ∪ T P_n=E_n\cup T P n = E n ∪ T P n P_n P n Q a \mathbb{Q}_a Q a G n G_n G n P n P_n P n g n = Q a ( G n ) g_n=\mathbb{Q}_a(G_n) g n = Q a ( G n )
Every boundary visit of the remaining picture has positive residence time. Its Poisson marks almost surely avoid the finitely many prescribed changes; seam completions preserve sites; and exposed histories avoid the boundary, so forced transports cannot cause an instantaneous boundary excursion. Consequently
1 G n ↑ 1 G P , g n ↑ g P > 0. \mathbf{1}_{G_n}\mathrel{\uparrow}\mathbf{1}_{G_P}, \qquad g_n \mathrel{\uparrow} g_P>0. 1 G n ↑ 1 G P , g n ↑ g P > 0. For all sufficiently large n n n I = P n I=P_n I = P n A n A_n A n P n P_n P n i i i T T T
Q a ( G n ∩ A n ) g n = Pr w a l k a , i ( τ T + < τ E n ) . (22) \frac{\mathbb{Q}_a(G_n\cap A_n)}{g_n}
=
\overset{\mathrm{walk}}{\Pr}_{\mathbf{a},i}\left(\tau_T^+<\tau_{E_n}\right). \tag*{(22)} g n Q a ( G n ∩ A n ) = Pr walk a , i ( τ T + < τ E n ) . (22) No assertion about the possibly singular earlier n n n
In a fixed remaining picture the first positive return from i i i T T T 1 A n \mathbf{1}_{A_n} 1 A n 1 R i ( E , T ) \mathbf{1}_{\mathcal{R}_i(E,T)} 1 R i ( E , T ) Q a ( ⋅ ∣ G P ) \mathbb{Q}_a(\mathord{\cdot}\mid G_P) Q a ( ⋅ ∣ G P )
On the right use one uncensored fresh trajectory for all n n n τ T + < ∞ \tau_T^+<\infty τ T + < ∞ { τ T + < τ E } \{\tau_T^+<\tau_E\} { τ T + < τ E }
We have obtained a Markov return problem in a periodic environment whose missing slots are the exposed cycles. The environment retains its full pinned-law distribution, while the walk resamples its transitions on every traversal. The next section develops the deterministic heat-flow estimates needed to bound these returns.
Kernels and the geometry of available slots
The fresh walk moves through a graph whose available slots change with physical time. We need two deterministic ways to control it: a Nash inequality converts dissipation into diffusion, and flows to infinity bound the pairing of a source mass with a test function by its Dirichlet energy. The first source bound permits many sparse starting points; the second becomes sharper when a single source has already diffused. We obtain both bounds on a fixed collection of nearest-neighbor sublattices contained in the interaction graph. The walk itself always uses all interaction bonds.
Coset coordinates and the killed kernels
Choose linearly independent vectors a 1 , a 2 , a 3 a_1,a_2,a_3 a 1 , a 2 , a 3 J ( a j ) > 0 J(a_j)>0 J ( a j ) > 0 A A A J J J Z 3 \mathbb{Z}^3 Z 3 A Z 3 A\mathbb{Z}^3 A Z 3 ∣ det A ∣ |\det A| ∣ det A ∣ ( u γ ) γ ∈ C (u_\gamma)_{\gamma\in\mathcal{C}} ( u γ ) γ ∈ C
x = u γ + A x ˉ , γ ∈ C , x ˉ ∈ Z 3 . (23) x=u_\gamma+A\bar{x}, \qquad\gamma\in\mathcal{C},\quad\bar{x}\in\mathbb{Z}^3. \tag*{(23)} x = u γ + A x ˉ , γ ∈ C , x ˉ ∈ Z 3 . (23) For z ∈ Z 3 z\in\mathbb{Z}^3 z ∈ Z 3 R ≥ 0 R\geq0 R ≥ 0
B R ( z ) = { w ∈ Z 3 : ∣ w − z ∣ ∞ ≤ R } , B R ( z ) = { x ∈ Z 3 : x ˉ ∈ B R ( z ) } . B_R(z)=\{w\in\mathbb{Z}^3:|w-z|_\infty\leq R\}, \qquad\mathcal{B}_R(z)=\{x\in\mathbb{Z}^3:\bar{x}\in B_R(z)\}. B R ( z ) = { w ∈ Z 3 : ∣ w − z ∣ ∞ ≤ R } , B R ( z ) = { x ∈ Z 3 : x ˉ ∈ B R ( z )} . Thus B R ( z ) \mathcal{B}_{R}(z) B R ( z ) x ˉ \bar{x} x ˉ ± a j \pm a_j ± a j B ∗ ≥ 1 B_{*} \ge1 B ∗ ≥ 1
J ( y − x ) > 0 ⟹ ∣ y ˉ − x ˉ ∣ ∞ ≤ B ∗ . (24) J(y-x)>0 \quad\Longrightarrow\quad\lvert\bar{y}-\bar{x}\rvert_{\infty}\le B_{*}. \tag*{(24)} J ( y − x ) > 0 ⟹ ∣ y ˉ − x ˉ ∣ ∞ ≤ B ∗ . (24) Indeed, y ˉ − x ˉ = A − 1 ( y − x + u γ − u γ ′ ) \bar{y}-\bar{x}=A^{-1}(y-x+u_{\gamma}-u_{\gamma'}) y ˉ − x ˉ = A − 1 ( y − x + u γ − u γ ′ ) S S S J J J
Fix an exposure a \mathbf{a} a θ \theta θ U U U ∂ Λ \partial\Lambda ∂ Λ U v Uv U v v v v
U + = U , U − = U ∗ , Q = U + U ∗ 2 , A Q = I − Q . (25) U_{+}=U,\qquad U_{-}=U^{*},\qquad Q=\frac{U+U^{*}}{2},\qquad A_{Q}=I-Q. \tag*{(25)} U + = U , U − = U ∗ , Q = 2 U + U ∗ , A Q = I − Q . (25) The ordered kernel U U U Q Q Q
Every interval kernel is an average of partial permutation matrices: sample the simultaneous trajectories of all holes and retain precisely those trajectories that survive the killing. Consequently it is nonnegative, with every row and column sum at most one, and contracts counting 1 1 1 2 2 2 K K K
∑ i ∣ ( K v ) i ∣ 2 ≤ ∑ i ( ∑ j K i j ) ( ∑ j K i j ∣ v j ∣ 2 ) ≤ ∑ j ∣ v j ∣ 2 . \sum_i \lvert(Kv)_i\rvert^2
\le\sum_i\left(\sum_j K_{ij}\right)\left(\sum_j K_{ij}\lvert v_j\rvert^2\right)
\le\sum_j\lvert v_j\rvert^2. i ∑ ∣( K v ) i ∣ 2 ≤ i ∑ ( j ∑ K ij ) ( j ∑ K ij ∣ v j ∣ 2 ) ≤ j ∑ ∣ v j ∣ 2 . These properties hold for adjoints, products, and further spatial killing. In particular Q Q Q A Q A_Q A Q
A split of a traversal always uses matching hole layers. At a prescribed transport or seam we distinguish its two sides and assign the operation to exactly one of the adjoining factors. Thus a full forward traversal from the post-seam cut at zero includes its seam at the end. Reversing a traversal reverses this same ordered list. Time integrals ignore the finitely many instantaneous operations.
At time t t t ℓ \ell ℓ Λ \Lambda Λ H t \mathcal{H}_t H t v v v
D t ( v ) = ∑ e = { i , j } ∈ E ( H t ) c e ( v ( i ) − v ( j ) ) 2 , c e = 1 2 J ( x j − x i ) . (26) \mathcal{D}_t(v)=\sum_{e=\{i,j\}\in E(\mathcal{H}_t)}c_e\bigl(v(i)-v(j)\bigr)^2,\qquad c_e=\frac{1}{2}J(x_j-x_i). \tag*{(26)} D t ( v ) = e = { i , j } ∈ E ( H t ) ∑ c e ( v ( i ) − v ( j ) ) 2 , c e = 2 1 J ( x j − x i ) . (26) Each unoriented edge is counted once. A killed vector is extended by zero on killed slots, including the boundary and the exterior. On a smooth interval, in elapsed time u u u
d d u ∥ v ( u ) ∥ 2 2 = − 2 D t ( u ) ( v ( u ) ) . (27) \frac{d}{du}\lVert v(u)\rVert_2^2=-2\mathcal{D}_{t(u)}(v(u)). \tag*{(27)} d u d ∥ v ( u ) ∥ 2 2 = − 2 D t ( u ) ( v ( u )) . (27) To verify this, pair the symmetric jump generator with v v v − 2 c e ( v ( i ) − v ( j ) ) 2 -2c_e(v(i)-v(j))^2 − 2 c e ( v ( i ) − v ( j ) ) 2 Λ ∘ \Lambda^\circ Λ ∘
Regular anchors and local diffusion
For a coset γ \gamma γ H t γ \mathcal{H}^{\gamma}_{t} H t γ
D t γ ( v ) = 1 2 ∑ { i , j } ∈ E ( H t γ ) ( v ( i ) − v ( j ) ) 2 , c 0 = min 1 ≤ j ≤ 3 J ( a j ) > 0. \mathcal{D}^{\gamma}_{t}(v)=\frac{1}{2}\sum_{\{i,j\}\in E(\mathcal{H}^{\gamma}_{t})}(v(i)-v(j))^{2}, \qquad c_{0}=\min_{1\leq j\leq3}J(a_{j})>0. D t γ ( v ) = 2 1 { i , j } ∈ E ( H t γ ) ∑ ( v ( i ) − v ( j ) ) 2 , c 0 = 1 ≤ j ≤ 3 min J ( a j ) > 0. The graphs for different cosets have disjoint vertex sets, and their edges are a subset of the full interaction edges. Therefore
D t ( v ) ≥ c 0 ∑ γ ∈ C D t γ ( v ) . (28) \mathcal{D}_{t}(v)\geq c_{0}\sum_{\gamma\in\mathcal{C}}\mathcal{D}^{\gamma}_{t}(v). \tag*{(28)} D t ( v ) ≥ c 0 γ ∈ C ∑ D t γ ( v ) . (28) This inequality is the only place where the geometric subgraphs replace the full graph; in particular it changes constants, not the diffusion matrix that will determine Bloch’s coefficient.
Fix
α = 1 100 , ν = 1 300 , L ≥ max { 6 , ℓ ∣ C ∣ } . \alpha=\frac{1}{100}, \qquad\nu=\frac{1}{300}, \qquad L\geq\max\{6,\ell|\mathcal{C}|\}. α = 100 1 , ν = 300 1 , L ≥ max { 6 , ℓ ∣ C ∣ } . Definition 3.1 (Sparse collections). A finite collection of slot points with site coordinates ( x i ) (x_i) ( x i ) sparse if
# { i : x ˉ i ∈ B h ( z ) } ≤ L ( 1 + h ) α ( z ∈ Z 3 , h ≥ 0 ) . (29) \#\{i:\bar{x}_i\in B_h(z)\}\leq L(1+h)^{\alpha}\qquad(z\in\mathbb{Z}^{3},\ h\geq0). \tag*{(29)} # { i : x ˉ i ∈ B h ( z )} ≤ L ( 1 + h ) α ( z ∈ Z 3 , h ≥ 0 ) . (29) Repeated bar coordinates count with their multiplicities, including points from different cosets. The same definition applies to a list of centers in Z 3 \mathbb{Z}^{3} Z 3
In coset γ \gamma γ blocked if it has no available slot; denote this finite set by O t γ \mathcal{O}^{\gamma}_{t} O t γ w ∈ O t γ w\in\mathcal{O}^{\gamma}_{t} w ∈ O t γ C t γ ( w ) \mathcal{C}^{\gamma}_{t}(w) C t γ ( w )
rad w C t γ ( w ) = max v ∈ C t γ ( w ) ∣ v − w ∣ ∞ . \operatorname{rad}_{w}\mathcal{C}^{\gamma}_{t}(w)=\max_{v\in\mathcal{C}^{\gamma}_{t}(w)}|v-w|_{\infty}. rad w C t γ ( w ) = v ∈ C t γ ( w ) max ∣ v − w ∣ ∞ . Boundary sites and all sites outside Λ \Lambda Λ ℓ \ell ℓ
We recall the precise local-connectivity input from [15, Lemma 5.5]. For every fixed integer b ≥ 1 b\geq1 b ≥ 1 ρ ( 3 , b ) ≥ b + 1 \rho(3,b)\geq b+1 ρ ( 3 , b ) ≥ b + 1 u , v ∉ F ⊂ Z 3 u,v\notin F\subset\mathbb{Z}^{3} u , v ∈ / F ⊂ Z 3 ∣ u − v ∣ ∞ ≤ b |u-v|_{\infty}\leq b ∣ u − v ∣ ∞ ≤ b # ( F ∩ B ρ ( 3 , b ) ( u ) ) < 6 \#(F\cap B_{\rho(3,b)}(u))<6 # ( F ∩ B ρ ( 3 , b ) ( u )) < 6 u u u v v v F F F ( 2 ρ ( 3 , b ) + 1 ) 3 (2\rho(3,b)+1)^{3} ( 2 ρ ( 3 , b ) + 1 ) 3
M ∗ ≥ max { 48 , ρ ( 3 , 44 ) + 2 } . (30) M_{*}\geq\max\{48,\rho(3,44)+2\}. \tag*{(30)} M ∗ ≥ max { 48 , ρ ( 3 , 44 ) + 2 } . (30) The numbers 44 and 48 cover the bounded connections around components of at most five blocked sites. Fix a lower threshold R 0 R_{0} R 0 R 0 ≥ M ∗ + 47 R_{0}\geq M_{*}+47 R 0 ≥ M ∗ + 47
Definition 3.2 (Regular slice). A slice t t t regular for the anchor z z z R R R γ \gamma γ
for every w ∈ B 11 R ( z ) w\in B_{11R}(z) w ∈ B 11 R ( z ) # ( O t γ ∩ B M ∗ ( w ) ) < 6 \#(\mathcal{O}^{\gamma}_{t}\cap B_{M_{*}}(w))<6 # ( O t γ ∩ B M ∗ ( w )) < 6
for every w ∈ O t γ ∖ B 10 R ( z ) w\in\mathcal{O}^{\gamma}_{t}\setminus B_{10R}(z) w ∈ O t γ ∖ B 10 R ( z ) rad w C t γ ( w ) ≤ ( 1 + ∣ w − z ∣ ∞ ) ν \operatorname{rad}_{w}\mathcal{C}^{\gamma}_{t}(w)\leq(1+|w-z|_{\infty})^{\nu} rad w C t γ ( w ) ≤ ( 1 + ∣ w − z ∣ ∞ ) ν
The anchor itself need not be an available site.
These are precisely the regular-anchor conditions in [15, Definition 5.3], on each coordinate lattice. They are conditions on a single deterministic graph; no pin probability estimate is assumed in the next three propositions.
Proposition 3.3 (Local Nash inequality). Suppose R ≥ R 0 R \ge R_0 R ≥ R 0 and t t t is regular for z z z at radius R R R Every nonnegative function v v v on H t \mathcal{H}_t H t supported on slots above B R ( z ) \mathcal{B}_R(z) B R ( z ) satisfies
∥ v ∥ 2 10 / 3 ≤ C D t ( v ) ∥ v ∥ 1 4 / 3 . (31) \lVert v\rVert_2^{10/3} \le C D_t(v)\lVert v\rVert_1^{4/3}. \tag*{(31)} ∥ v ∥ 2 10/3 ≤ C D t ( v ) ∥ v ∥ 1 4/3 . (31) The constant is independent of R R R z z z t t t and the unavailable slots.
Proof. We give the transfer argument of [15 ] , Proposition 5.6, followed by the lattice averaging proof of Nash’s inequality; the energy-to-diffusion method originates in [14 ] . First work in one coset and write its bar coordinates simply as sites. At each unblocked site w w w v v v V ( w ) V(w) V ( w ) B R + 3 ( z ) B_{R+3}(z) B R + 3 ( z ) ϕ ( w ) \phi(w) ϕ ( w )
f ( w ) = { V ( w ) , w ∉ O t γ , V ( ϕ ( w ) ) , w ∈ O t γ ∩ B R + 3 ( z ) , 0 , w ∈ O t γ ∖ B R + 3 ( z ) . f(w) =
\begin{cases}
V(w), & w \notin\mathcal{O}_t^\gamma,\\
V(\phi(w)), & w \in\mathcal{O}_t^\gamma\cap B_{R+3}(z),\\
0, & w \in\mathcal{O}_t^\gamma\setminus B_{R+3}(z).
\end{cases} f ( w ) = ⎩ ⎨ ⎧ V ( w ) , V ( ϕ ( w )) , 0 , w ∈ / O t γ , w ∈ O t γ ∩ B R + 3 ( z ) , w ∈ O t γ ∖ B R + 3 ( z ) . Then f f f B R + 1 ( z ) B_{R+1}(z) B R + 1 ( z )
∥ v ∥ 2 2 ≤ ℓ ∥ f ∥ 2 2 , ∥ f ∥ 1 ≤ 7 ∥ v ∥ 1 . (32) \lVert v\rVert_2^2 \le\ell\lVert f\rVert_2^2,\qquad\lVert f\rVert_1 \le7\lVert v\rVert_1. \tag*{(32)} ∥ v ∥ 2 2 ≤ ℓ ∥ f ∥ 2 2 , ∥ f ∥ 1 ≤ 7 ∥ v ∥ 1 . (32) For a lattice edge with a nonzero endpoint value of f f f B R + 3 ( z ) B_{R+3}(z) B R + 3 ( z ) b = 44 b=44 b = 44
E Z 3 ( f ) : = ∑ { w , w ′ } : ∣ w − w ′ ∣ 1 = 1 ( f ( w ) − f ( w ′ ) ) 2 ≤ C D t γ ( v ) . (33) \mathcal{E}_{\mathbb{Z}^3}(f) := \sum_{\{w,w'\}:|w-w'|_1=1} (f(w)-f(w'))^2 \le C D_t^\gamma(v). \tag*{(33)} E Z 3 ( f ) := { w , w ′ } : ∣ w − w ′ ∣ 1 = 1 ∑ ( f ( w ) − f ( w ′ ) ) 2 ≤ C D t γ ( v ) . (33) For completeness, average f f f { 0 , … , b − 1 } 3 \{0,\ldots,b-1\}^3 { 0 , … , b − 1 } 3 A b f A_bf A b f ℓ 2 \ell^2 ℓ 2 ∥ f − A b f ∥ 2 2 ≤ C b 2 E Z 3 ( f ) \lVert f-A_bf\rVert_2^2 \le Cb^2\mathcal{E}_{\mathbb{Z}^3}(f) ∥ f − A b f ∥ 2 2 ≤ C b 2 E Z 3 ( f ) b − 3 / 2 b^{-3/2} b − 3/2 ∥ A b f ∥ 2 2 ≤ b − 3 ∥ f ∥ 1 2 \lVert A_bf\rVert_2^2 \le b^{-3}\lVert f\rVert_1^2 ∥ A b f ∥ 2 2 ≤ b − 3 ∥ f ∥ 1 2
∥ f ∥ 2 2 ≤ C b 2 E Z 3 ( f ) + 2 b − 3 ∥ f ∥ 1 2 . \lVert f\rVert_2^2 \le Cb^2\mathcal{E}_{\mathbb{Z}^3}(f)+2b^{-3}\lVert f\rVert_1^2. ∥ f ∥ 2 2 ≤ C b 2 E Z 3 ( f ) + 2 b − 3 ∥ f ∥ 1 2 . For f ≠ 0 f\ne0 f = 0 b = ⌈ ( 4 ∥ f ∥ 1 2 / ∥ f ∥ 2 2 ) 1 / 3 ⌉ b=\lceil(4\lVert f\rVert_1^2/\lVert f\rVert_2^2)^{1/3}\rceil b = ⌈( 4 ∥ f ∥ 1 2 / ∥ f ∥ 2 2 ) 1/3 ⌉ ∥ f ∥ 1 ≥ ∥ f ∥ 2 \lVert f\rVert_1\ge\lVert f\rVert_2 ∥ f ∥ 1 ≥ ∥ f ∥ 2 ∥ f ∥ 2 10 / 3 ≤ C E Z 3 ( f ) ∥ f ∥ 1 4 / 3 \lVert f\rVert_2^{10/3}\le C\mathcal{E}_{\mathbb{Z}^3}(f)\lVert f\rVert_1^{4/3} ∥ f ∥ 2 10/3 ≤ C E Z 3 ( f ) ∥ f ∥ 1 4/3 (32) –(33) give (31) for one coset with D t γ D_t^\gamma D t γ v γ v_\gamma v γ γ \gamma γ
( ∑ γ ∥ v γ ∥ 2 2 ) 5 / 3 ≤ ∣ C ∣ 2 / 3 ∑ γ ∥ v γ ∥ 2 10 / 3 , ∥ v γ ∥ 1 ≤ ∥ v ∥ 1 , \left(\sum_\gamma\lVert v_\gamma\rVert_2^2\right)^{5/3}
\le|\mathcal{C}|^{2/3}\sum_\gamma\lVert v_\gamma\rVert_2^{10/3},
\qquad
\lVert v_\gamma\rVert_1 \le\lVert v\rVert_1, ( γ ∑ ∥ v γ ∥ 2 2 ) 5/3 ≤ ∣ C ∣ 2/3 γ ∑ ∥ v γ ∥ 2 10/3 , ∥ v γ ∥ 1 ≤ ∥ v ∥ 1 , and (3.6) to obtain the assertion on all cosets.
Flows and sparse sources
The Nash inequality controls a mass while it evolves. We next control its pairing with an arbitrary function at a single slice. A flow with divergence equal to that mass gives precisely this bound by summation by parts. Fix an orientation ( e − , e + ) (e^{-},e^{+}) ( e − , e + ) F F F Y Y Y
⟨ div F , Y ⟩ = ∑ e F ( e ) ( Y ( e − ) − Y ( e + ) ) . (34) \langle\operatorname{div} F,Y\rangle=\sum_{e}F(e)\bigl(Y(e^{-})-Y(e^{+})\bigr). \tag*{(34)} ⟨ div F , Y ⟩ = e ∑ F ( e ) ( Y ( e − ) − Y ( e + ) ) . (34) For an edge in one coset and a coordinate site y y y dist ( e , y ) \operatorname{dist}(e,y) dist ( e , y ) y y y e e e
We state the deterministic input, including the features of its construction needed for a sharper estimate below. Its hypotheses do not refer to an exposure law or to a relation between R R R
Lemma 3.4 (Routed unit flows: geometric input). Consider an available-slot nearest-neighbor graph on Z 3 \mathbb{Z}^{3} Z 3 ℓ \ell ℓ z z z R ≥ R 0 R\ge R_{0} R ≥ R 0 a a a y ∈ B R ( z ) y\in B_{R}(z) y ∈ B R ( z ) F a F_{a} F a
div F a = 1 { a } , ∣ F a ( e ) ∣ ≤ C ( 1 + dist ( e , y ) ) − k , k = 2 − 3 ν = 199 100 . (35) \operatorname{div}F_{a}=\mathbf{1}_{\{a\}},\qquad|F_{a}(e)|\le C\bigl(1+\operatorname{dist}(e,y)\bigr)^{-k},\qquad k=2-3\nu=\frac{199}{100}. \tag*{(35)} div F a = 1 { a } , ∣ F a ( e ) ∣ ≤ C ( 1 + dist ( e , y ) ) − k , k = 2 − 3 ν = 100 199 . (35) The flows can be chosen with the following common routing construction. For each blocked component C C C H C H_{C} H C Box C + \operatorname{Box}_{C}^{+} Box C +
There are simple surviving routes for the original lattice edges. Each route is contained in its original edge together with the boxes Box C + \operatorname{Box}_{C}^{+} Box C + F a F_{a} F a a a a G y G_{y} G y
∣ G y ( e ′ ) ∣ ≤ C ( 1 + dist ( e ′ , y ) ) − 2 . (36) |G_{y}(e')|\le C\bigl(1+\operatorname{dist}(e',y)\bigr)^{-2}. \tag*{(36)} ∣ G y ( e ′ ) ∣ ≤ C ( 1 + dist ( e ′ , y ) ) − 2 . (36) All constants are uniform over the graph, anchor, radius, and source.
This is [15 ] , Proposition 6.2 together with its stated construction: hull containment and surviving boundary routes are in [15 ] , Lemma 6.1 and Section 6.1; the once-applied endpoint map and the routes, including adjacent maximal hulls, follow Equations (6.2) –(65) there; the bounded connector, lattice flow, and lifting are in the proof of Proposition 6.2, Equations (68) –(70) . These statements allow arbitrary available-slot counts between one and ℓ \ell ℓ
Two such flows, whose source sites are y , y ′ y,y' y , y ′
∑ e ∣ F a ( e ) F b ( e ) ∣ ≤ C ( 1 + ∣ y − y ′ ∣ ∞ ) − q , q = 2 k − 3 = 49 50 . (37) \sum_{e}|F_{a}(e)F_{b}(e)|\le C\bigl(1+|y-y'|_{\infty}\bigr)^{-q},\qquad q=2k-3=\frac{49}{50}. \tag*{(37)} e ∑ ∣ F a ( e ) F b ( e ) ∣ ≤ C ( 1 + ∣ y − y ′ ∣ ∞ ) − q , q = 2 k − 3 = 50 49 . (37) Here the flows may use different regular anchors. To check this estimate, put H = 1 + ∣ y − y ′ ∣ ∞ H=1+|y-y'|_\infty H = 1 + ∣ y − y ′ ∣ ∞ ∣ y − y ′ ∣ ∞ / 3 |y-y'|_\infty/3 ∣ y − y ′ ∣ ∞ /3 C H − k CH^{-k} C H − k C H 3 − k CH^{3-k} C H 3 − k H H H C ∑ j ≥ 0 ( 2 j H ) 3 − 2 k C\sum_{j\ge0}(2^jH)^{3-2k} C ∑ j ≥ 0 ( 2 j H ) 3 − 2 k C H 3 − 2 k CH^{3-2k} C H 3 − 2 k 3 / 2 < k < 3 3/2<k<3 3/2 < k < 3 H H H [15 ] , Lemma 6.3.
Proposition 3.5 (Energy of sparse sources). Let R ≥ R 0 R\ge R_0 R ≥ R 0 ( z i ) i = 1 m (z_i)_{i=1}^m ( z i ) i = 1 m I ⊆ { 1 , … , m } I\subseteq\{1,\ldots,m\} I ⊆ { 1 , … , m } z i z_i z i R R R i ∈ I i\in I i ∈ I μ i ≥ 0 \mu_i\ge0 μ i ≥ 0 B R ( z i ) \mathcal{B}_R(z_i) B R ( z i ) Y Y Y
∣ ⟨ ∑ i ∈ I μ i , Y ⟩ ∣ 2 ≤ C m R α D t ( Y ) . (38) \left|\left\langle\sum_{i\in I}\mu_i,Y\right\rangle\right|^2\le CmR^\alpha\mathcal{D}_t(Y). \tag*{(38)} ⟨ i ∈ I ∑ μ i , Y ⟩ 2 ≤ C m R α D t ( Y ) . (38) Proof. In each coset sum the unit flows of Lemma 3.4 with the coefficients supplied by the restrictions of the μ i \mu_i μ i F γ F^\gamma F γ ∣ z i − z j ∣ ∞ > 4 R |z_i-z_j|_\infty>4R ∣ z i − z j ∣ ∞ > 4 R ∣ z i − z j ∣ ∞ / 2 |z_i-z_j|_\infty/2 ∣ z i − z j ∣ ∞ /2 (37) and the mass bound at each center give
∑ e ∣ F γ ( e ) ∣ 2 ≤ C ∑ i ∈ I [ # { j ∈ I : ∣ z j − z i ∣ ∞ ≤ 4 R } + ∑ j ∈ I ∣ z j − z i ∣ ∞ > 4 R ( 1 + ∣ z j − z i ∣ ∞ ) − q ] . \sum_e|F^\gamma(e)|^2\le C\sum_{i\in I}\left[\#\{j\in I:|z_j-z_i|_\infty\le4R\}+\sum_{\substack{j\in I\\|z_j-z_i|_\infty>4R}}(1+|z_j-z_i|_\infty)^{-q}\right]. e ∑ ∣ F γ ( e ) ∣ 2 ≤ C i ∈ I ∑ # { j ∈ I : ∣ z j − z i ∣ ∞ ≤ 4 R } + j ∈ I ∣ z j − z i ∣ ∞ > 4 R ∑ ( 1 + ∣ z j − z i ∣ ∞ ) − q . The first count is at most C L R α CLR^\alpha C L R α 2 n 4 R < ∣ z i − z j ∣ ∞ ≤ 2 n + 1 4 R 2^n4R<|z_i-z_j|_\infty\le2^{n+1}4R 2 n 4 R < ∣ z i − z j ∣ ∞ ≤ 2 n + 1 4 R C L ∑ n ≥ 0 ( 2 n R ) α − q ≤ C CL\sum_{n\ge0}(2^nR)^{\alpha-q}\le C C L ∑ n ≥ 0 ( 2 n R ) α − q ≤ C q > α q>\alpha q > α R ≥ 1 R\ge1 R ≥ 1 ∑ γ ∑ e ∣ F γ ( e ) ∣ 2 ≤ C m R α \sum_\gamma\sum_e|F^\gamma(e)|^2\le CmR^\alpha ∑ γ ∑ e ∣ F γ ( e ) ∣ 2 ≤ C m R α (34) , Cauchy–Schwarz on the disjoint coset edge sets, and (28) , we get
∣ ⟨ ∑ i ∈ I μ i , Y ⟩ ∣ 2 ≤ C m R α ∑ γ D t γ ( Y ) ≤ C m R α D t ( Y ) . \left|\left\langle\sum_{i\in I}\mu_i,Y\right\rangle\right|^2\le CmR^\alpha\sum_\gamma\mathcal{D}_t^\gamma(Y)\le CmR^\alpha\mathcal{D}_t(Y). ⟨ i ∈ I ∑ μ i , Y ⟩ 2 ≤ C m R α γ ∑ D t γ ( Y ) ≤ C m R α D t ( Y ) . This includes the empty set I I I
The sharper bound for a spread source
The next estimate converts a 2-norm bound C 1 τ − 3 / 4 C_1\tau^{-3/4} C 1 τ − 3/4 C τ − 1 / 2 C\tau^{-1/2} C τ − 1/2 k < 2 k<2 k < 2 (35) alone loses a power of τ \tau τ
Lemma 3.6 (Decay up to the localization radius). The same flows as in Lemma 3.4 satisfy
∣ F a ( e ) ∣ ≤ C ( 1 + h ) − 2 if h = dist ( e , y ) ≤ R . (39) |F_a(e)|\le C(1+h)^{-2}\qquad\text{if }h=\operatorname{dist}(e,y)\le R. \tag*{(39)} ∣ F a ( e ) ∣ ≤ C ( 1 + h ) − 2 if h = dist ( e , y ) ≤ R . (39) The constant and the lower threshold R 0 R_0 R 0 R R R
Proof. A blocked component meeting B 10 R ( z ) B_{10R}(z) B 10 R ( z ) M ∗ ≥ 48 M_*\ge48 M ∗ ≥ 48 B o x C \mathrm{Box}_C Box C B 10 R ( z ) B_{10R}(z) B 10 R ( z ) b C = min w ∈ C ∣ w − z ∣ ∞ > 10 R b_C=\min_{w\in C}|w-z|_\infty>10R b C = min w ∈ C ∣ w − z ∣ ∞ > 10 R
diam ∞ ( Box C + ) ≤ 2 ( 1 + b C ) ν + 10 ≤ b C / 10 \operatorname{diam}_\infty(\operatorname{Box}^{+}_C)\le2(1+b_C)^\nu+10\le b_C/10 diam ∞ ( Box C + ) ≤ 2 ( 1 + b C ) ν + 10 ≤ b C /10 after increasing the fixed threshold R 0 R_0 R 0 B 9 R ( z ) B_{9R}(z) B 9 R ( z )
Now let e e e h ≤ R h\le R h ≤ R B 2 R + 1 ( z ) B_{2R+1}(z) B 2 R + 1 ( z ) e ′ e' e ′ e e e e ′ e' e ′ e ′ e' e ′ e ′ e' e ′ e e e e e e h h h e e e
Proposition 3.7 (Spread-source energy). Suppose R ≥ R 0 R\ge R_0 R ≥ R 0 1 ≤ τ ≤ R 1\le\tau\le R 1 ≤ τ ≤ R t t t z z z R R R μ ≥ 0 \mu\ge0 μ ≥ 0 B R ( z ) B_R(z) B R ( z )
∥ μ ∥ 1 ≤ 1 , ∥ μ ∥ 2 ≤ C 1 τ − 3 / 4 . \lVert\mu\rVert_1\le1,\qquad\lVert\mu\rVert_2\le C_1\tau^{-3/4}. ∥ μ ∥ 1 ≤ 1 , ∥ μ ∥ 2 ≤ C 1 τ − 3/4 . For every finitely supported real function Y Y Y H t \mathcal{H}_t H t
∣ ⟨ μ , Y ⟩ ∣ 2 ≤ C τ − 1 / 2 D t ( Y ) . (40) |\langle\mu,Y\rangle|^2\le C\tau^{-1/2}D_t(Y). \tag*{(40)} ∣ ⟨ μ , Y ⟩ ∣ 2 ≤ C τ − 1/2 D t ( Y ) . (40) The constant may depend on C 1 C_1 C 1 τ \tau τ R R R
Proof. Work first on one coset, with the restriction of μ \mu μ F = ∑ a μ ( a ) F a F=\sum_a\mu(a)F_a F = ∑ a μ ( a ) F a div F = μ \operatorname{div}F=\mu div F = μ y a y_a y a s = τ s=\sqrt{\tau} s = τ
Index each edge by one endpoint site, with bounded multiplicity, and put μ ˉ ( w ) = ∑ a : y a = w μ ( a ) \bar{\mu}(w)=\sum_{a:y_a=w}\mu(a) μ ˉ ( w ) = ∑ a : y a = w μ ( a ) ℓ \ell ℓ ∥ μ ˉ ∥ 2 ≤ ℓ ∥ μ ∥ 2 \lVert\bar{\mu}\rVert_2\le\sqrt{\ell}\lVert\mu\rVert_2 ∥ μ ˉ ∥ 2 ≤ ℓ ∥ μ ∥ 2 s ≤ R s\le R s ≤ R
κ s ( w ) = C ( 1 + ∣ w ∣ ∞ ) − 2 1 { ∣ w ∣ ∞ ≤ s + 1 } , ∥ κ s ∥ 1 ≤ C s . \kappa_s(w)=C(1+|w|_\infty)^{-2}\mathbf{1}_{\{|w|_\infty\le s+1\}},\qquad\lVert\kappa_s\rVert_1\le Cs. κ s ( w ) = C ( 1 + ∣ w ∣ ∞ ) − 2 1 { ∣ w ∣ ∞ ≤ s + 1 } , ∥ κ s ∥ 1 ≤ C s . Here changing the minimum distance to an edge into the distance from its indexed endpoint costs at most one. Young’s inequality yields
∥ F n e a r ∥ 2 ≤ C ∥ κ s ∗ μ ˉ ∥ 2 ≤ C s ∥ μ ∥ 2 ≤ C τ − 1 / 4 . (41) \lVert F_{\mathrm{near}}\rVert_2\le C\lVert\kappa_s*\bar{\mu}\rVert_2\le Cs\lVert\mu\rVert_2\le C\tau^{-1/4}. \tag*{(41)} ∥ F near ∥ 2 ≤ C ∥ κ s ∗ μ ˉ ∥ 2 ≤ C s ∥ μ ∥ 2 ≤ C τ − 1/4 . (41) For the far part of each individual flow, count at most C ( 1 + h ) 2 C(1+h)^2 C ( 1 + h ) 2 h h h R R R k k k
∑ e : dist ( e , y a ) > s ∣ F a ( e ) ∣ 2 ≤ C ∑ s < h ≤ R ( 1 + h ) − 2 + C ∑ h > R ( 1 + h ) 2 − 2 k ≤ C ( τ − 1 / 2 + R − 49 / 50 ) ≤ C τ − 1 / 2 . (42) \sum_{e:\,\operatorname{dist}(e,y_a)>s}|F_a(e)|^2
\le C\sum_{s<h\le R}(1+h)^{-2}+C\sum_{h>R}(1+h)^{2-2k}
\le C(\tau^{-1/2}+R^{-49/50})\le C\tau^{-1/2}. \tag*{(42)} e : dist ( e , y a ) > s ∑ ∣ F a ( e ) ∣ 2 ≤ C s < h ≤ R ∑ ( 1 + h ) − 2 + C h > R ∑ ( 1 + h ) 2 − 2 k ≤ C ( τ − 1/2 + R − 49/50 ) ≤ C τ − 1/2 . (42) The last comparison uses exactly R ≥ τ ≥ 1 R\ge\tau\ge1 R ≥ τ ≥ 1 49 / 50 > 1 / 2 49/50>1/2 49/50 > 1/2 ℓ 2 \ell^2 ℓ 2 ∑ a μ ( a ) ≤ 1 \sum_a\mu(a)\le1 ∑ a μ ( a ) ≤ 1 ∥ F f a r ∥ 2 ≤ C τ − 1 / 4 \lVert F_{\mathrm{far}}\rVert_2\le C\tau^{-1/4} ∥ F far ∥ 2 ≤ C τ − 1/4 ∑ e ∣ F ( e ) ∣ 2 ≤ C τ − 1 / 2 \sum_e|F(e)|^2\le C\tau^{-1/2} ∑ e ∣ F ( e ) ∣ 2 ≤ C τ − 1/2
Apply this construction on every coset. Each restriction satisfies the same mass and 2-norm bounds, so the sum of all coset flow energies is at most C ∣ E ∣ τ − 1 / 2 C|\mathcal{E}|\tau^{-1/2} C ∣ E ∣ τ − 1/2
In particular, when τ = K β \tau= K\beta τ = K β R = ⌈ a ( K + 1 ) β ⌉ R = \lceil a(K+1)\beta\rceil R = ⌈ a ( K + 1 ) β ⌉ a ≥ 1 a \ge1 a ≥ 1 K K K
We now prove that a prescribed sparse set of slot points is unlikely to belong to cycles avoiding the pins. This supplies the regular slices needed for diffusion. The argument is a finite downward induction on the pin set: the estimate for a larger pin set controls the geometry used to remove some of those pins. It adapts the induction and obstacle estimates of [15 ] , with the interaction rates and the coset coordinates retained throughout.
All constants in this section may depend on the fixed parameters of Section 3, but not on the cube, pin sets, time grid, cut, spatial anchor, or inverse temperature. Dependence on an integer K K K o K ( a ( β ) ) o_K(a(\beta)) o K ( a ( β )) K K K a ( β ) a(\beta) a ( β ) β → ∞ \beta\to\infty β → ∞ K K K
Blocking probabilities from an induction hypothesis
Set p = β − 9 / 10 p = \beta^{-9/10} p = β − 9/10 β ≥ 1 \beta\ge1 β ≥ 1 G ⊂ R / β Z \mathcal{G} \subset\mathbb{R}/\beta\mathbb{Z} G ⊂ R / β Z p 12 p^{12} p 12 V G = V Λ × G \mathcal{V}_{\mathcal{G}} = V_{\Lambda} \times\mathcal{G} V G = V Λ × G P = D 0 ∪ F P = D_0 \cup F P = D 0 ∪ F F ⊆ V G F \subseteq\mathcal{V}_{\mathcal{G}} F ⊆ V G I ( P ) \mathrm{I}(P) I ( P )
P P ( T ′ ⊆ B P ) ≤ p ∣ T ′ ∣ (43) \mathbb{P}_{P}\left(T' \subseteq B_P\right) \le p^{|T'|} \tag*{(43)} P P ( T ′ ⊆ B P ) ≤ p ∣ T ′ ∣ (43) for every nonempty sparse T ′ T' T ′ P P P
Here and below sparseness counts slot multiplicities as in Definition 3.1. Until the final subsection we assume I ( P ) \mathrm{I}(P) I ( P )
Expose B P B_P B P P P \mathbb{P}_P P P γ \gamma γ O t γ O_t^\gamma O t γ q q q A A A
P P ( A ⊆ O q γ ) ≤ p ∣ A ∣ . (44) \mathbb{P}_P\left(A \subseteq O_q^\gamma\right) \le p^{|A|}. \tag*{(44)} P P ( A ⊆ O q γ ) ≤ p ∣ A ∣ . (44) To see this, select a fixed slot at each corresponding physical site. If any selected slot is pinned, or any site lies outside the cube, the blocking event is impossible. Otherwise the selected slot points form a permissible test in (4.1).
Lemma 4.1 (Blocking at deterministic times). Assume I ( P ) \mathrm{I}(P) I ( P ) t t t A A A
P P ( A ⊆ O t γ ) ≤ C p 6 \mathbb{P}_P\left(A \subseteq O_t^\gamma\right) \le C p^6 P P ( A ⊆ O t γ ) ≤ C p 6 if ∣ A ∣ = 6 |A| = 6 ∣ A ∣ = 6 {#eq:4.3}
P P ( A ⊆ O t γ ) ≤ ( C p ) ∣ A ∣ \mathbb{P}_P\left(A \subseteq O_t^\gamma\right) \le(C\sqrt{p})^{|A|} P P ( A ⊆ O t γ ) ≤ ( C p ) ∣ A ∣ if A ≠ ∅ A \ne\varnothing A = ∅ {#eq:4.4}
The constants are uniform in the choice and size of the grid.
Proof. Let q q q t t t q = t q=t q = t t ∈ G t \in\mathcal{G} t ∈ G ( q , t ] (q,t] ( q , t ] p 12 p^{12} p 12 t t t q q q S , J S,J S , J C p 12 Cp^{12} C p 12 L ≥ 6 L \ge6 L ≥ 6 p 6 + C p 12 ≤ C p 6 p^6 + Cp^{12} \le Cp^6 p 6 + C p 12 ≤ C p 6
For a sparse set of n ≥ 1 n \ge1 n ≥ 1 ⌈ n / 2 ⌉ \lceil n/2\rceil ⌈ n /2 ⌉ q q q 2 n p ⌈ n / 2 ⌉ 2^n p^{\lceil n/2\rceil} 2 n p ⌈ n /2 ⌉ j = ⌈ n / 4 ⌉ j=\lceil n/4\rceil j = ⌈ n /4 ⌉ N N N
P P ( N ≥ j ) ≤ E P ( N j ) ≤ ( C n p 12 ) j j ! ≤ ( C ′ p 12 ) j . \mathbb{P}_P(N \ge j) \le\mathbb{E}_P \binom{N}{j} \le\frac{(Cnp^{12})^j}{j!} \le(C'p^{12})^j. P P ( N ≥ j ) ≤ E P ( j N ) ≤ j ! ( C n p 12 ) j ≤ ( C ′ p 12 ) j . Here n / j ≤ 4 n/j \le4 n / j ≤ 4 j ! ≥ ( j / e ) j j! \ge(j/e)^j j ! ≥ ( j / e ) j ( C p ) n (C\sqrt{p})^n ( C p ) n C C C
Sparse witnesses for large blocked components
We next turn the joint blocking bound into a tail for a connected obstacle. Components C t γ ( w ) C_t^\gamma(w) C t γ ( w ) rad w C t γ ( w ) = max v ∈ C t γ ( w ) ∣ v − w ∣ ∞ \operatorname{rad}_w C_t^\gamma(w)=\max_{v\in C_t^\gamma(w)}|v-w|_\infty rad w C t γ ( w ) = max v ∈ C t γ ( w ) ∣ v − w ∣ ∞ [15 ] ; we include the sparseness and counting arguments because only sparse joint tests are available.
Lemma 4.2 (Component tail). There are β 0 , c , C , λ > 0 \beta_0,c,C,\lambda>0 β 0 , c , C , λ > 0 β ≥ β 0 \beta\ge\beta_0 β ≥ β 0 I ( P ) \mathcal{I}(P) I ( P )
P P ( w ∈ O t γ , rad w C t γ ( w ) ≥ h ) ≤ C exp ( − c h λ ) , h ≥ 1. (45) \mathbb{P}_P\left(w \in\mathcal{O}_t^\gamma,\operatorname{rad}_w C_t^\gamma(w) \ge h\right) \le C\exp(-ch^\lambda), \qquad h \ge1. \tag*{(45)} P P ( w ∈ O t γ , rad w C t γ ( w ) ≥ h ) ≤ C exp ( − c h λ ) , h ≥ 1. (45) This holds for every deterministic time t t t γ \gamma γ w w w β ≥ β 0 \beta\ge\beta_0 β ≥ β 0
Proof. Fix an integer b > 100 b>100 b > 100 b α > 4 b^\alpha>4 b α > 4 h j = 16 b j h_j=16b^j h j = 16 b j j ≥ 0 j \ge0 j ≥ 0 j j j w w w x v x_v x v j j j x ∅ = w x_\varnothing=w x ∅ = w k ≥ 1 k \ge1 k ≥ 1 h k + 8 h_k+8 h k + 8 h k / 4 h_k/4 h k /4
A blocked path from w w w h j h_j h j j ≥ 1 j \ge1 j ≥ 1 h j h_j h j w w w h j / 2 h_j/2 h j /2 w w w h j / 2 + 8 h_j/2+8 h j /2 + 8 h j / 2 − 8 h_j/2-8 h j /2 − 8 h j − 1 h_{j-1} h j − 1 h j − 1 + 8 h_{j-1}+8 h j − 1 + 8
h j / 2 − 2 ( h j − 1 + 8 ) ≥ h j / 4. h_j/2-2(h_{j-1}+8) \ge h_j/4. h j /2 − 2 ( h j − 1 + 8 ) ≥ h j /4. Recursing inside these crossings constructs the children and keeps all their descendant leaves in the indicated balls.
Every admissible witness has 2 j 2^j 2 j h < h k / 8 h<h_k/8 h < h k /8 h k h_k h k 2 k − 1 2^{k-1} 2 k − 1 h k − 1 / 8 ≤ h < h k / 8 h_{k-1}/8 \le h < h_k/8 h k − 1 /8 ≤ h < h k /8 1 ≤ k ≤ j 1 \le k \le j 1 ≤ k ≤ j
L ( 1 + h ) α ≥ L ( 2 b k − 1 ) α ≥ 2 k − 1 . L(1+h)^\alpha\ge L(2b^{k-1})^\alpha\ge2^{k-1}. L ( 1 + h ) α ≥ L ( 2 b k − 1 ) α ≥ 2 k − 1 . Below h 0 / 8 h_0/8 h 0 /8 h j / 8 h_j/8 h j /8 2 j 2^j 2 j
Let A j A_j A j
A 0 = 1 , A j ≤ C h j 3 A j − 1 2 , 2 − j log A j ≤ ∑ k = 1 j 2 − k log ( C h k 3 ) ≤ log C b . A_0 = 1,\qquad A_j \le C h_j^3 A_{j-1}^2,\qquad2^{-j}\log A_j \le\sum_{k=1}^{j} 2^{-k}\log(C h_k^3) \le\log C_b. A 0 = 1 , A j ≤ C h j 3 A j − 1 2 , 2 − j log A j ≤ k = 1 ∑ j 2 − k log ( C h k 3 ) ≤ log C b . Thus A j ≤ C b 2 j A_j \le C_b^{2^j} A j ≤ C b 2 j ( C p ) 2 j (C\sqrt{p})^{2^j} ( C p ) 2 j β 0 \beta_0 β 0 C b C p ≤ e − 2 C_b C\sqrt{p}\le e^{-2} C b C p ≤ e − 2 β ≥ β 0 \beta\ge\beta_0 β ≥ β 0
P ( w ∈ O t γ , rad w C t γ ( w ) ≥ h j ) ≤ e − 2 ⋅ 2 j . \mathbb{P}\left(w\in\mathcal{O}_t^\gamma,\operatorname{rad}_w\mathcal{C}_t^\gamma(w)\ge h_j\right)\le e^{-2\cdot2^j}. P ( w ∈ O t γ , rad w C t γ ( w ) ≥ h j ) ≤ e − 2 ⋅ 2 j . For h j ≤ h < h j + 1 h_j\le h<h_{j+1} h j ≤ h < h j + 1 2 j ≥ ( h / ( 16 b ) ) λ 2^j\ge(h/(16b))^\lambda 2 j ≥ ( h / ( 16 b ) ) λ λ = log b 2 > 0 \lambda=\log_b 2>0 λ = log b 2 > 0 C C C 1 ≤ h < 16 1\le h<16 1 ≤ h < 16 (45) .
Proposition 4.3 (Probability of an irregular slice). Under I(P), for β ≥ β 0 \beta\ge\beta_0 β ≥ β 0 t t t z ∈ Z 3 z\in\mathbb{Z}^3 z ∈ Z 3 R ≥ 1 R\ge1 R ≥ 1
P ( t is not regular for z at radius R ) ≤ δ R : = C R 3 p 6 + C exp ( − c R ξ ) , ξ = ν λ > 0. (46) \mathbb{P}(t\text{ is not regular for }z\text{ at radius }R)\le\delta_R:=CR^3p^6+C\exp(-cR^\xi),\qquad\xi=\nu\lambda>0. \tag*{(46)} P ( t is not regular for z at radius R ) ≤ δ R := C R 3 p 6 + C exp ( − c R ξ ) , ξ = ν λ > 0. (46) Proof. For each coset there are O ( R 3 ) O(R^3) O ( R 3 ) B 11 R ( z ) B_{11R}(z) B 11 R ( z ) M ∗ M_* M ∗ eq:4.3 and a union bound give C R 3 p 6 CR^3p^6 C R 3 p 6
C ∑ w : ∣ w − z ∣ ∞ > 10 R e − c ( 1 + ∣ w − z ∣ ∞ ) ν ≤ C e − c ′ R ν λ . C\sum_{w:\lvert w-z\rvert_\infty>10R}e^{-c(1+\lvert w-z\rvert_\infty)^\nu}\le Ce^{-c'R^{\nu\lambda}}. C w : ∣ w − z ∣ ∞ > 10 R ∑ e − c ( 1 + ∣ w − z ∣ ∞ ) ν ≤ C e − c ′ R ν λ . For the last inequality, count at most C ( 1 + n ) 2 C(1+n)^2 C ( 1 + n ) 2 n n n
We have obtained regularity at prescribed times using only the induction hypothesis for P P P K ≥ 1 K\ge1 K ≥ 1
τ = K β , R = R K = ⌈ a ( K + 1 ) β ⌉ . (47) \tau=K\beta,\qquad R=R_K=\lceil a(K+1)\beta\rceil. \tag*{(47)} τ = K β , R = R K = ⌈ a ( K + 1 ) β ⌉ . (47) The constant a ≥ 1 a\ge1 a ≥ 1 K K K a ≥ R 0 a\ge R_0 a ≥ R 0
δ R K = O K ( β − 12 / 5 ) = o K ( β − 3 / 2 ) . (48) \delta_{R_K}=O_K(\beta^{-12/5})=o_K(\beta^{-3/2}). \tag*{(48)} δ R K = O K ( β − 12/5 ) = o K ( β − 3/2 ) . (48) Call the exposure good for z , R z,R z , R β / 100 \beta/100 β /100 G z , R G_{z,R} G z , R
P ( G z , R c ) ≤ 100 δ R . (49) \mathbb{P}(G_{z,R}^{c})\le100\delta_R. \tag*{(49)} P ( G z , R c ) ≤ 100 δ R . (49) Neither this assertion nor Proposition 4.3 requires independence between time slices.
Localization and diffusion
To localize at z z z B R ( z ) \mathcal{B}_R(z) B R ( z ) U U U U d , R U_{d,R} U d , R d ∈ { + , − } d \in\{+,-\} d ∈ { + , − } U − , R = U + , R ∗ U_{-,R}=U_{+,R}^{*} U − , R = U + , R ∗
Lemma 4.4 (Localization). One may choose a a a K K K I ( P ) \mathcal{I}(P) I ( P ) i i i P P P z = x ˉ i z=\bar{x}_i z = x ˉ i K \mathcal{K} K 2 K + 2 2K+2 2 K + 2 i i i K R \mathcal{K}_R K R z z z
E P ∥ ( K − K R ) 1 { i } ∥ 1 ≤ C K e − c β . (50) \mathbb{E}_{P}\left\lVert(\mathcal{K}-\mathcal{K}_R)\mathbf{1}_{\{i\}}\right\rVert_1 \le C_K e^{-c\beta}. \tag*{(50)} E P ( K − K R ) 1 { i } 1 ≤ C K e − c β . (50) The same estimate holds for localization of the walk on the full slot lattice, with all slots available and no exposed trajectories.
Proof. Sample the actual picture under P P \mathbb{P}_P P P B R ( z ) \mathcal{B}_R(z) B R ( z )
An exit requires at least R / B ∗ R/B_* R / B ∗
j = ⌈ R B ∗ ( 2 K + 2 ) ⌉ ≥ a β 2 B ∗ − 1 j=\left\lceil\frac{R}{B_*(2K+2)}\right\rceil\ge\frac{a\beta}{2B_*}-1 j = ⌈ B ∗ ( 2 K + 2 ) R ⌉ ≥ 2 B ∗ a β − 1 steps, starting at a site in B R ( z ) \mathcal{B}_R(z) B R ( z )
For a fixed starting site, ignore slot compatibility to overcount all lists of j j j β j / j ! \beta^j/j! β j / j !
( C β ) j j ! . \frac{(C\beta)^j}{j!}. j ! ( C β ) j . Choose a a a j ≥ a β / ( 3 B ∗ ) j\ge a\beta/(3B_*) j ≥ a β / ( 3 B ∗ ) β \beta β e − c β e^{-c\beta} e − c β j ! ≥ ( j / e ) j j!\ge(j/e)^j j ! ≥ ( j / e ) j 2 K + 2 2K+2 2 K + 2 O ( R 3 ) O(R^3) O ( R 3 ) c c c C K C_K C K
All factorial estimates were applied unconditionally under P P \mathbb{P}_P P P
Proposition 4.5 (Diffusion on a good exposure). Fix an exposure good for z z z R K R_K R K z z z b ′ ≥ β / 16 b' \ge\beta/16 b ′ ≥ β /16
∥ v o u t ∥ 2 ≤ C ( b ′ ) − 3 / 4 . (51) \lVert v_{\mathrm{out}}\rVert_2 \le C(b')^{-3/4}. \tag*{(51)} ∥ v out ∥ 2 ≤ C ( b ′ ) − 3/4 . (51) The same statement holds for the adjoint of any such subinterval. For such a kernel K R K_R K R b ′ ≥ β / 8 b' \ge\beta/8 b ′ ≥ β /8
max j , k ( K R ) j k ≤ C ( b ′ ) − 3 / 2 . (52) \max_{j,k}(K_R)_{jk} \le C(b')^{-3/2}. \tag*{(52)} j , k max ( K R ) j k ≤ C ( b ′ ) − 3/2 . (52) The constants are independent of K K K R K R_K R K
Proof. This is Nash’s dissipation argument, applied only during regular times; see [14 ] , pp. 935–936. Let q = ∥ v ∥ 2 q=\lVert v\rVert_2 q = ∥ v ∥ 2
q ′ ≤ − c q 5 / 3 , ( q − 2 / 3 ) ′ ≥ c ′ > 0 when q > 0. q' \le-cq^{5/3}, \qquad(q^{-2/3})' \ge c' > 0 \quad\text{when } q>0. q ′ ≤ − c q 5/3 , ( q − 2/3 ) ′ ≥ c ′ > 0 when q > 0. At other smooth times and at instantaneous operations the norm does not increase. If it becomes zero, the conclusion is immediate. The input and output masses are at most one, by contraction. The subinterval has only two possible partial end pieces; all intermediate pieces have length β \beta β ( b ′ / β + 2 ) β / 100 (b'/\beta+2)\beta/100 ( b ′ / β + 2 ) β /100 b ′ ≥ β / 16 b' \ge\beta/16 b ′ ≥ β /16 67 b ′ / 100 67b'/100 67 b ′ /100 (51) . The same count holds after reversing the subinterval, and the reversed instantaneous factors are contractions. For (52) , split at the temporal midpoint: an entry is the inner product of a forward mass from its starting slot and an adjoint mass from its ending slot. Each propagates for b ′ / 2 ≥ β / 16 b'/2 \ge\beta/16 b ′ /2 ≥ β /16 (51) give the result. □
The hitting energy and the induction step
We have proved diffusion under the geometry supplied by I ( P ) I(P) I ( P ) E E E P = E ∪ T P=E\cup T P = E ∪ T
Lemma 4.6 (Hitting energy). Let E E E T ≠ ∅ T\ne\varnothing T = ∅ θ \theta θ E E E P = E ∪ T P=E\cup T P = E ∪ T m T = ∣ T ∣ m_T=|T| m T = ∣ T ∣ η = 1 T \eta=1_T η = 1 T P P \mathbb{P}_P P P f = 1 f=1 f = 1 T T T f f f T T T 0 ≤ f ≤ 1 0\le f\le1 0 ≤ f ≤ 1 f f f g = f − η g=f-\eta g = f − η
∑ i ∈ T q i ( a ) ≤ m T − ⟨ f , A Q f ⟩ = ⟨ η , Q η ⟩ + 2 ⟨ Q η , g ⟩ − ⟨ g , A Q g ⟩ . (53) \sum_{i\in T}q_i(a)\le m_T-\langle f,A_Qf\rangle
=\langle\eta,Q\eta\rangle+2\langle Q\eta,g\rangle-\langle g,A_Qg\rangle. \tag*{(53)} i ∈ T ∑ q i ( a ) ≤ m T − ⟨ f , A Q f ⟩ = ⟨ η , Q η ⟩ + 2 ⟨ Q η , g ⟩ − ⟨ g , A Q g ⟩ . (53) Proof. As in [15 ] , Lemma 4.5, construct f f f f 0 = η f_0=\eta f 0 = η f n + 1 = 1 f_{n+1}=1 f n + 1 = 1 T T T f n + 1 = U ∗ f n f_{n+1}=U^*f_n f n + 1 = U ∗ f n T T T 0 ≤ f n ≤ f n + 1 ≤ 1 0\le f_n\le f_{n+1}\le1 0 ≤ f n ≤ f n + 1 ≤ 1 U ∗ f = f U^*f=f U ∗ f = f T T T
Since T T T T T T E ∖ D 0 E\setminus D_0 E ∖ D 0 q i ( a ) ≤ ( U ∗ f ) i q_i(a)\le(U^*f)_i q i ( a ) ≤ ( U ∗ f ) i i ∈ T i\in T i ∈ T T T T f = 1 f=1 f = 1 T T T
⟨ f , A Q f ⟩ = ⟨ ( I − U ∗ ) f , f ⟩ = m T − ∑ i ∈ T ( U ∗ f ) i . \langle f,A_Qf\rangle=\langle(I-U^*)f,f\rangle
=m_T-\sum_{i\in T}(U^*f)_i. ⟨ f , A Q f ⟩ = ⟨( I − U ∗ ) f , f ⟩ = m T − i ∈ T ∑ ( U ∗ f ) i . The quadratic forms of U U U U ∗ U^* U ∗ Q Q Q ⟨ η , g ⟩ = 0 \langle\eta,g\rangle=0 ⟨ η , g ⟩ = 0 f = η + g f=\eta+g f = η + g (53) .
For an arbitrary real vector y y y E y = ⟨ y , A Q y ⟩ E_y=\langle y,A_Qy\rangle E y = ⟨ y , A Q y ⟩ d ∈ { + , − } d\in\{+,-\} d ∈ { + , − } u ∈ I = [ β / 3 , 2 β / 3 ] u\in I=[\beta/3,2\beta/3] u ∈ I = [ β /3 , 2 β /3 ]
U d = B d ( u ) C d ( u ) , Y d ( u ) = B d ( u ) ∗ y . U_d=B_d(u)C_d(u), \qquad Y_d(u)=B_d(u)^*y. U d = B d ( u ) C d ( u ) , Y d ( u ) = B d ( u ) ∗ y . Here C d ( u ) C_d(u) C d ( u ) B d ( u ) B_d(u) B d ( u ) t d ( u ) = θ + u t_d(u)=\theta+u t d ( u ) = θ + u θ − u \theta-u θ − u β \beta β
Lemma 4.7 (Suffix energy). With these definitions and zero extension to the available-slot graph at the split,
∫ I D t d ( u ) ( Y d ( u ) ) d u ≤ 1 2 ( ∥ y ∥ 2 2 − ∥ U d ∗ y ∥ 2 2 ) ≤ E y . (54) \int_I D_{t_d(u)}(Y_d(u))\,du\le\frac{1}{2}\left(\lVert y\rVert_2^2-\lVert U_d^*y\rVert_2^2\right)\le E_y. \tag*{(54)} ∫ I D t d ( u ) ( Y d ( u )) d u ≤ 2 1 ( ∥ y ∥ 2 2 − ∥ U d ∗ y ∥ 2 2 ) ≤ E y . (54) Moreover ∥ Y d ( u ) ∥ ∞ ≤ ∥ y ∥ ∞ \lVert Y_d(u)\rVert_\infty\le\lVert y\rVert_\infty ∥ Y d ( u ) ∥ ∞ ≤ ∥ y ∥ ∞
Proof. As u u u B d ( u ) ∗ y B_d(u)^*y B d ( u ) ∗ y I I I
E y − 1 2 ( ∥ y ∥ 2 2 − ∥ U d ∗ y ∥ 2 2 ) = 1 2 ∥ y − U d ∗ y ∥ 2 2 ≥ 0 , E_y-\frac{1}{2}\left(\lVert y\rVert_2^2-\lVert U_d^*y\rVert_2^2\right)=\frac{1}{2}\lVert y-U_d^*y\rVert_2^2\ge0, E y − 2 1 ( ∥ y ∥ 2 2 − ∥ U d ∗ y ∥ 2 2 ) = 2 1 ∥ y − U d ∗ y ∥ 2 2 ≥ 0 , since ⟨ y , U d ∗ y ⟩ = ⟨ y , Q y ⟩ \langle y,U_d^*y\rangle=\langle y,Qy\rangle ⟨ y , U d ∗ y ⟩ = ⟨ y , Q y ⟩
Theorem 4.8 (Uniform sparse-pin estimate). There is β 1 < ∞ \beta_1<\infty β 1 < ∞ independent of the cube and pins, such that for every β ≥ β 1 \beta\ge\beta_1 β ≥ β 1 every allowed pin set E E E and every nonempty sparse set T T T at one deterministic time, disjoint from E E E
P E ( T ⊆ B E ) ≤ β − 9 ∣ T ∣ / 10 . (55) \mathbb{P}_E(T\subseteq B_E)\le\beta^{-9|T|/10}. \tag*{(55)} P E ( T ⊆ B E ) ≤ β − 9∣ T ∣/10 . (55) Consequently (46) and (49) hold under every allowed pin law, with no induction hypothesis.
Proof. First fix a grid G \mathcal{G} G I ( E ) \mathrm{I}(E) I ( E ) D 0 D_0 D 0 E E E T T T E E E P = E ∪ T P=E\cup T P = E ∪ T I ( P ) \mathrm{I}(P) I ( P ) P P \mathbb{P}_P P P K = 1 K=1 K = 1 R = R 1 R=R_1 R = R 1
The first term in (53) obeys
E P ⟨ η , Q η ⟩ ≤ C m T ( e − c β + δ R + R α β − 3 / 2 ) . (56) \mathbb{E}_P\langle\eta,Q\eta\rangle\le C m_T\left(e^{-c\beta}+\delta_R+R^\alpha\beta^{-3/2}\right). \tag*{(56)} E P ⟨ η , Q η ⟩ ≤ C m T ( e − c β + δ R + R α β − 3/2 ) . (56) Indeed, propagate a unit mass from each i ∈ T i\in T i ∈ T z i = x ˉ i z_i=\bar{x}_i z i = x ˉ i C e − c β Ce^{-c\beta} C e − c β z i , R z_i,R z i , R C β − 3 / 2 C\beta^{-3/2} C β − 3/2 L ( 1 + R ) α L(1+R)^\alpha L ( 1 + R ) α T T T 100 δ R 100\delta_R 100 δ R i i i E g = ⟨ g , A Q g ⟩ E_g=\langle g,AQg\rangle E g = ⟨ g , A Q g ⟩ y = g y=g y = g C d , R , z i ( u ) C_{d,R,z_i}(u) C d , R , z i ( u ) z i z_i z i
M d ( u ) = ∑ i ∈ T 1 { t d ( u ) r e g u l a r f o r z i a t r a d i u s R } C d , R , z i ( u ) 1 { i } , F d = 1 ∣ I ∣ ∫ I ⟨ M d ( u ) , Y d ( u ) ⟩ d u . M_d(u)=\sum_{i\in T}\mathbf{1}_{\{t_{d}(u)\ {\rm regular\ for}\ z_i\ {\rm at\ radius}\ R\}}C_{d,R,z_i}(u)\mathbf{1}_{\{i\}},\qquad F_d=\frac{1}{|I|}\int_I\langle M_d(u),Y_d(u)\rangle\,du. M d ( u ) = i ∈ T ∑ 1 { t d ( u ) regular for z i at radius R } C d , R , z i ( u ) 1 { i } , F d = ∣ I ∣ 1 ∫ I ⟨ M d ( u ) , Y d ( u )⟩ d u . The unchanged pairing ⟨ U d η , g ⟩ = ⟨ C d ( u ) η , Y d ( u ) ⟩ \langle U_d\eta,g\rangle=\langle C_d(u)\eta,Y_d(u)\rangle ⟨ U d η , g ⟩ = ⟨ C d ( u ) η , Y d ( u )⟩ u u u 0 ≤ g ≤ 1 0\leq g\leq1 0 ≤ g ≤ 1
E P ∣ ⟨ U d η , g ⟩ − F d ∣ ≤ C m T ( e − c β + δ R ) . (57) \mathbb{E}_P\left|\langle U_d\eta,g\rangle-F_d\right|\leq Cm_T\left(e^{-c\beta}+\delta_R\right). \tag*{(57)} E P ∣ ⟨ U d η , g ⟩ − F d ∣ ≤ C m T ( e − c β + δ R ) . (57) No good-exposure indicator is needed here: the next estimate uses only regularity at the split slice. Each retained summand of M d ( u ) M_d(u) M d ( u ) B R ( z i ) \mathcal{B}_R(z_i) B R ( z i ) T T T u u u
∣ F d ∣ ≤ ( C m T R α ∣ I ∣ E g ) 1 / 2 . |F_d|\leq\left(\frac{Cm_TR^\alpha}{|I|}E_g\right)^{1/2}. ∣ F d ∣ ≤ ( ∣ I ∣ C m T R α E g ) 1/2 . Because ∣ I ∣ = β / 3 |I|=\beta/3 ∣ I ∣ = β /3 E g ≥ 0 E_g\geq0 E g ≥ 0 E g \sqrt{E_g} E g
F + + F − − E g ≤ C m T R α β . F_++F_--E_g\leq\frac{Cm_TR^\alpha}{\beta}. F + + F − − E g ≤ β C m T R α . Now 2 ⟨ Q η , g ⟩ = ⟨ U + η , g ⟩ + ⟨ U − η , g ⟩ 2\langle Q\eta,g\rangle=\langle U_+\eta,g\rangle+\langle U_-\eta,g\rangle 2 ⟨ Q η , g ⟩ = ⟨ U + η , g ⟩ + ⟨ U − η , g ⟩
r ( E , T ) ≤ C ( e − c β + δ R + R α β − 3 / 2 + R α β − 1 ) = o ( β − 9 / 10 ) . (58) r(E,T)\leq C\left(e^{-c\beta}+\delta_R+R^\alpha\beta^{-3/2}+R^\alpha\beta^{-1}\right)=o(\beta^{-9/10}). \tag*{(58)} r ( E , T ) ≤ C ( e − c β + δ R + R α β − 3/2 + R α β − 1 ) = o ( β − 9/10 ) . (58) Here R = ⌈ 2 a β ⌉ R=\lceil2a\beta\rceil R = ⌈ 2 a β ⌉ O ( β − 99 / 100 ) O(\beta^{-99/100}) O ( β − 99/100 ) β 1 \beta_1 β 1 2 r ( E , T ) ≤ p 2r(E,T)\leq p 2 r ( E , T ) ≤ p P E ( T ⊆ B E ) ≤ ( 2 r ( E , T ) ) m T ≤ p m T \mathbb{P}_E(T\subseteq B_E)\leq(2r(E,T))^{m_T}\leq p^{m_T} P E ( T ⊆ B E ) ≤ ( 2 r ( E , T ) ) m T ≤ p m T
For arbitrary allowed E E E p 12 p^{12} p 12 P P P I ( P ) I(P) I ( P )
All time integrals above are of measurable functions of the exposure. The histories have finitely many jumps, their transition kernels are measurable, and the hitting function is an increasing limit of finite-horizon probabilities. The sharp one-slot estimate of order β − 3 / 2 \beta^{-3/2} β − 3/2
The long-return remainder
The sparse-pin estimate has made diffusion available under every allowed pin law. We now improve its one-slot consequence to the order β − 3 / 2 \beta^{-3/2} β − 3/2 2 K − 1 2K-1 2 K − 1 C K − 1 / 2 β − 3 / 2 + o K ( β − 3 / 2 ) CK^{-1/2}\beta^{-3/2}+o_K(\beta^{-3/2}) C K − 1/2 β − 3/2 + o K ( β − 3/2 ) C C C K K K K K K β \beta β
Fix an allowed pin set E E E i ∉ E i\notin E i ∈ / E θ \theta θ x x x z = x ˉ z=\bar{x} z = x ˉ P = E ∪ { i } P=E\cup\{i\} P = E ∪ { i } P P \mathbb{P}_P P P U U U Q = ( U + U ∗ ) / 2 Q=(U+U^*)/2 Q = ( U + U ∗ ) /2 A Q = I − Q A_Q=I-Q A Q = I − Q η = 1 { i } \eta=\mathbf{1}_{\{i\}} η = 1 { i } f f f { i } \{i\} { i } f ( i ) = 1 f(i)=1 f ( i ) = 1 0 ≤ f ≤ 1 0\le f\le1 0 ≤ f ≤ 1 f = U ∗ f f=U^*f f = U ∗ f i i i g = f − η g=f-\eta g = f − η K ≥ 1 K\ge1 K ≥ 1 R = R K R=R_K R = R K τ = K β \tau=K\beta τ = K β o K o_K o K
Lemma 5.1 (Finite return expansion). For almost every exposure, define
v = ∑ n = 1 K − 1 Q n η , y = g − v , E y = ⟨ y , A Q y ⟩ , v=\sum_{n=1}^{K-1}Q^n\eta,\qquad y=g-v,\qquad E_y=\langle y,A_Qy\rangle, v = n = 1 ∑ K − 1 Q n η , y = g − v , E y = ⟨ y , A Q y ⟩ , with the empty sum interpreted as zero. Then E y ≥ 0 E_y\ge0 E y ≥ 0 ∥ y ∥ ∞ ≤ K + 1 \lVert y\rVert_\infty\le K+1 ∥ y ∥ ∞ ≤ K + 1
q i ≤ ∑ n = 1 2 K − 1 ⟨ η , Q n η ⟩ + 2 ⟨ Q K η , y ⟩ − E y . q_i\le\sum_{n=1}^{2K-1}\langle\eta,Q^n\eta\rangle+2\langle Q^K\eta,y\rangle-E_y. q i ≤ n = 1 ∑ 2 K − 1 ⟨ η , Q n η ⟩ + 2 ⟨ Q K η , y ⟩ − E y . Proof. Lemma 4.6 gives
q i ≤ Q i i + 2 ⟨ Q η , g ⟩ − ⟨ g , A Q g ⟩ . q_i\le Q_{ii}+2\langle Q\eta,g\rangle-\langle g,A_Qg\rangle. q i ≤ Q ii + 2 ⟨ Q η , g ⟩ − ⟨ g , A Q g ⟩ . Every Q n η Q^n\eta Q n η ∥ v ∥ ∞ ≤ K − 1 \lVert v\rVert_\infty\le K-1 ∥ v ∥ ∞ ≤ K − 1 ∥ y ∥ ∞ ≤ K \lVert y\rVert_\infty\le K ∥ y ∥ ∞ ≤ K Q Q Q E y ≥ 0 E_y\ge0 E y ≥ 0 A Q v = Q η − Q K η A_Qv=Q\eta-Q^K\eta A Q v = Q η − Q K η g = v + y g=v+y g = v + y 2 ⟨ Q K η , y ⟩ 2\langle Q^K\eta,y\rangle 2 ⟨ Q K η , y ⟩ a n = ⟨ η , Q n η ⟩ a_n=\langle\eta,Q^n\eta\rangle a n = ⟨ η , Q n η ⟩
Q i i + 2 ⟨ Q η , v ⟩ − ⟨ v , A Q v ⟩ = a 1 + ⟨ v , Q η ⟩ + ⟨ v , Q K η ⟩ = a 1 + ∑ n = 1 K − 1 a n + 1 + ∑ n = 1 K − 1 a n + K = ∑ n = 1 2 K − 1 a n . \begin{aligned}
Q_{ii}+2\langle Q\eta,v\rangle-\langle v,A_Qv\rangle
=a_1+\langle v,Q\eta\rangle+\langle v,Q^K\eta\rangle\\
&=a_1+\sum_{n=1}^{K-1}a_{n+1}+\sum_{n=1}^{K-1}a_{n+K}
=\sum_{n=1}^{2K-1}a_n.
\end{aligned} Q ii + 2 ⟨ Q η , v ⟩ − ⟨ v , A Q v ⟩ = a 1 + ⟨ v , Q η ⟩ + ⟨ v , Q K η ⟩ = a 1 + n = 1 ∑ K − 1 a n + 1 + n = 1 ∑ K − 1 a n + K = n = 1 ∑ 2 K − 1 a n . This proves (5.1). No infinite Green series or assumption of eventual boundary absorption is used.
Proposition 5.2 (Long-return bound). For every fixed integer K ≥ 1 K\ge1 K ≥ 1 β → ∞ \beta\to\infty β → ∞
E P [ 2 ⟨ Q K η , y ⟩ − E y ] ≤ C K − 1 / 2 β − 3 / 2 + o K ( β − 3 / 2 ) . (59) \mathbb{E}_{P}\bigl[2\langle Q^K\eta,y\rangle-E_y\bigr]
\le CK^{-1/2}\beta^{-3/2}+o_K(\beta^{-3/2}). \tag*{(59)} E P [ 2 ⟨ Q K η , y ⟩ − E y ] ≤ C K − 1/2 β − 3/2 + o K ( β − 3/2 ) . (59) Here C C C K K K
Proof. Set w = Q K − 1 η w=Q^{K-1}\eta w = Q K − 1 η I = [ β / 3 , 2 β / 3 ] I=[\beta/3,2\beta/3] I = [ β /3 , 2 β /3 ]
2 ⟨ Q K η , y ⟩ = ⟨ U + w , y ⟩ + ⟨ U − w , y ⟩ . 2\langle Q^K\eta,y\rangle=\langle U+w,y\rangle+\langle U-w,y\rangle. 2 ⟨ Q K η , y ⟩ = ⟨ U + w , y ⟩ + ⟨ U − w , y ⟩ . For d ∈ { + , − } d\in\{+,-\} d ∈ { + , − } u ∈ I u\in I u ∈ I
U d = B d ( u ) C d ( u ) , Y d ( u ) = B d ( u ) ∗ y . U_d=B_d(u)C_d(u),\qquad Y_d(u)=B_d(u)^*y. U d = B d ( u ) C d ( u ) , Y d ( u ) = B d ( u ) ∗ y . Both vectors in the resulting pairing
⟨ U d w , y ⟩ = ⟨ C d ( u ) w , Y d ( u ) ⟩ \langle U_dw,y\rangle=\langle C_d(u)w,Y_d(u)\rangle ⟨ U d w , y ⟩ = ⟨ C d ( u ) w , Y d ( u )⟩ are on the same hole layer at physical time t d ( u ) = θ + u t_d(u)=\theta+u t d ( u ) = θ + u θ − u \theta-u θ − u β \beta β
∥ Y d ( u ) ∥ ∞ ≤ K + 1 , ∫ I D t d ( u ) ( Y d ( u ) ) d u ≤ E y . (60) \lVert Y_d(u)\rVert_\infty\le K+1,\qquad\int_I \mathcal{D}_{t_d(u)}(Y_d(u))\,du\le E_y. \tag*{(60)} ∥ Y d ( u ) ∥ ∞ ≤ K + 1 , ∫ I D t d ( u ) ( Y d ( u )) d u ≤ E y . (60) In particular the energy on the right is the original one-period quadratic form, with no localization error.
Localize the mass in the first argument throughout its propagation from i i i
Q R = 1 2 ( U + , R + U − , R ) , μ d ( u ) = C d , R ( u ) Q R K − 1 η . Q_R=\frac{1}{2}(U_{+,R}+U_{-,R}),\qquad\mu_d(u)=C_{d,R}(u)Q_R^{K-1}\eta. Q R = 2 1 ( U + , R + U − , R ) , μ d ( u ) = C d , R ( u ) Q R K − 1 η . where the subscript R R R B R ( z ) \mathcal{B}_R(z) B R ( z ) Q Q Q Q R Q_R Q R
E P ∥ C d ( u ) w − μ d ( u ) ∥ 1 ≤ C K e − c β , u ∈ I . (61) \mathbb{E}_P\lVert C_d(u)w-\mu_d(u)\rVert_1\le CK e^{-c\beta},\qquad u\in I. \tag*{(61)} E P ∥ C d ( u ) w − μ d ( u ) ∥ 1 ≤ C K e − c β , u ∈ I . (61) Let G G G z , R z,R z , R
F d = 1 ∣ I ∣ ∫ I 1 G 1 { t d ( u ) is regular for z , R } ⟨ μ d ( u ) , Y d ( u ) ⟩ d u . F_d=\frac{1}{|I|}\int_I \mathbf{1}_G\mathbf{1}_{\{t_d(u)\text{ is regular for }z,R\}}\langle\mu_d(u),Y_d(u)\rangle\,du. F d = ∣ I ∣ 1 ∫ I 1 G 1 { t d ( u ) is regular for z , R } ⟨ μ d ( u ) , Y d ( u )⟩ d u . The original pairing is independent of u u u μ d ( u ) \mu_d(u) μ d ( u )
E P ∣ ⟨ U d w , y ⟩ − F d ∣ ≤ ( K + 1 ) ( C K e − c β + P P ( G c ) + 1 ∣ I ∣ ∫ I P P ( t d ( u ) is not regular for z , R ) d u ) = o K ( β − 3 / 2 ) . (62) \mathbb{E}_P\left|\langle U_dw,y\rangle-F_d\right|
\le(K+1)\left(CK e^{-c\beta}+\mathbb{P}_P(G^c)+\frac{1}{|I|}\int_I\mathbb{P}_P(t_d(u)\text{ is not regular for }z,R)\,du\right)
=o_K(\beta^{-3/2}). \tag*{(62)} E P ∣ ⟨ U d w , y ⟩ − F d ∣ ≤ ( K + 1 ) ( C K e − c β + P P ( G c ) + ∣ I ∣ 1 ∫ I P P ( t d ( u ) is not regular for z , R ) d u ) = o K ( β − 3/2 ) . (62) These are unconditional expectations under the exposure law. In particular no independence between the propagated mass, the suffix, and the regularity events is asserted or needed.
On a retained slice, every component of μ d ( u ) \mu_d(u) μ d ( u )
b = ( K − 1 ) β + u ≥ ( K − 2 3 ) β ≥ 1 3 K β . b=(K-1)\beta+u\ge\left(K-\frac{2}{3}\right)\beta\ge\frac{1}{3}K\beta. b = ( K − 1 ) β + u ≥ ( K − 3 2 ) β ≥ 3 1 K β . Proposition 4.5 and convexity of the norm give ∥ μ d ( u ) ∥ 2 ≤ C τ − 3 / 4 \lVert\mu_d(u)\rVert_2\le C\tau^{-3/4} ∥ μ d ( u ) ∥ 2 ≤ C τ − 3/4 K K K B R ( z ) \mathcal{B}_R(z) B R ( z ) 1 ≤ τ ≤ R 1\le\tau\le R 1 ≤ τ ≤ R C s p C_{\mathrm{sp}} C sp K K K
∣ ⟨ μ d ( u ) , Y d ( u ) ⟩ ∣ ≤ ( C s p τ − 1 / 2 D t d ( u ) ( Y d ( u ) ) ) 1 / 2 . |\langle\mu_d(u),Y_d(u)\rangle|\le\left(C_{\mathrm{sp}}\tau^{-1/2}\mathcal{D}_{t_d(u)}(Y_d(u))\right)^{1/2}. ∣ ⟨ μ d ( u ) , Y d ( u )⟩ ∣ ≤ ( C sp τ − 1/2 D t d ( u ) ( Y d ( u )) ) 1/2 . Cauchy–Schwarz in u u u
∣ F d ∣ ≤ ( 3 C s p τ − 1 / 2 β ) 1 / 2 E y 1 / 2 . |F_d|\le\left(\frac{3C_{\mathrm{sp}}\tau^{-1/2}}{\beta}\right)^{1/2}E_y^{1/2}. ∣ F d ∣ ≤ ( β 3 C sp τ − 1/2 ) 1/2 E y 1/2 . This also holds on G c G^c G c F d = 0 F_d=0 F d = 0
F + + F − − E y ≤ 3 C s p τ − 1 / 2 β = 3 C s p K − 1 / 2 β − 3 / 2 . F_++F_--E_y\le\frac{3C_{\mathrm{sp}}\tau^{-1/2}}{\beta}=3C_{\mathrm{sp}}K^{-1/2}\beta^{-3/2}. F + + F − − E y ≤ β 3 C sp τ − 1/2 = 3 C sp K − 1/2 β − 3/2 . Take expectations and add the two errors in (5.5). This proves (5.2) and identifies a leading constant that is independent of K K K
Corollary 5.3 (Uniform singleton sparsity). For every sufficiently large β \beta β uniformly in all allowed pin sets E E E cubes, cuts, and slot-time points i ∉ E i \notin E i ∈ / E
r ( E , { i } ) ≤ C β − 3 / 2 . r(E,\{i\}) \le C\beta^{-3/2}. r ( E , { i }) ≤ C β − 3/2 . Consequently, for every allowed pin set D D D and every slot-time point j j j at a deterministic time,
P D ( j ∈ B D ) ≤ C β − 3 / 2 . (63) \mathbb{P}_{D}(j \in B_D) \le C\beta^{-3/2}. \tag*{(63)} P D ( j ∈ B D ) ≤ C β − 3/2 . (63) Proof. Take K = 1 K=1 K = 1
r ( E , { i } ) = E P q i ≤ E P Q i i + C β − 3 / 2 + o ( β − 3 / 2 ) . r(E,\{i\})=\mathbb{E}_{P}q_i\le\mathbb{E}_{P}Q_{ii}+C\beta^{-3/2}+o(\beta^{-3/2}). r ( E , { i }) = E P q i ≤ E P Q ii + C β − 3/2 + o ( β − 3/2 ) . The two one-turn localization losses are at most C e − c β Ce^{-c\beta} C e − c β C β − 3 / 2 C\beta^{-3/2} C β − 3/2 o ( β − 3 / 2 ) o(\beta^{-3/2}) o ( β − 3/2 ) (49) . This bounds E P Q i i \mathbb{E}_{P}Q_{ii} E P Q ii (5.6) . For j ∉ D j\notin D j ∈ / D E = D E=D E = D j j j
P D ( j ∈ B D ) = 2 d D ( j ) = 2 r ( D , { j } ) 1 + r ( D , { j } ) ≤ 2 r ( D , { j } ) . \mathbb{P}_{D}(j\in B_D)=2d_D(j)=\frac{2r(D,\{j\})}{1+r(D,\{j\})}\le2r(D,\{j\}). P D ( j ∈ B D ) = 2 d D ( j ) = 1 + r ( D , { j }) 2 r ( D , { j }) ≤ 2 r ( D , { j }) . For j ∈ D j\in D j ∈ D
The estimate (63) is obtained only after closing the sparse-pin induction. It will now control encounters between exposed strands and an independent ideal path, allowing the explicit finite sum in (5.1) to be evaluated sharply.
Finite returns and the Bloch coefficient
We identify the finite return terms using the walk with every slot available. Its site motion uses all interaction bonds, so its diffusion matrix is D S , J D_{S,J} D S , J o K ( 1 ) o_K(1) o K ( 1 ) O ( β − 3 / 2 ) O(\beta^{-3/2}) O ( β − 3/2 )
Fix K K K E E E i ∉ E i\notin E i ∈ / E P = E ∪ { i } P=E\cup\{i\} P = E ∪ { i } R = R K R=R_K R = R K x x x i i i z = x ˉ z=\bar{x} z = x ˉ d = ( d 1 , … , d n ) ∈ { + , − } n \mathbf{d}=(d_1,\ldots,d_n)\in\{+,-\}^n d = ( d 1 , … , d n ) ∈ { + , − } n d 1 d_1 d 1
W ( d ) = U d n ⋯ U d 1 , W R ( d ) = U d n , R ⋯ U d 1 , R . W(\mathbf{d})=U_{d_n}\cdots U_{d_1},\qquad W_R(\mathbf{d})=U_{d_n,R}\cdots U_{d_1,R}. W ( d ) = U d n ⋯ U d 1 , W R ( d ) = U d n , R ⋯ U d 1 , R . The factors are formed in one fixed exposure; each traversal then uses fresh transition randomness. The ideal kernels have all slots available, exchange rate J ( y − x ) / 2 J(y-x)/2 J ( y − x ) /2 Z 3 \mathbb{Z}^3 Z 3 W ^ ∞ ( d ) \widehat{W}_{\infty}(\mathbf{d}) W ∞ ( d ) Λ ∘ \Lambda^\circ Λ ∘ B R ( z ) B_R(z) B R ( z ) W ^ R ( d ) \widehat{W}_{R}(\mathbf{d}) W R ( d )
The full interaction kernel
The site of a full ideal path jumps by v v v ℓ J ( v ) / 2 = S J ( v ) \ell J(v)/2 = SJ(v) ℓ J ( v ) /2 = S J ( v ) λ J = S ∑ v J ( v ) \lambda_J = S\sum_v J(v) λ J = S ∑ v J ( v )
ϕ ( k ) = S ∑ v J ( v ) ( 1 − cos ( k ⋅ v ) ) , κ = 1 ( 4 π ) 3 / 2 det D S , J , \phi(k) = S\sum_v J(v)(1-\cos(k\cdot v)), \qquad\kappa= \frac{1}{(4\pi)^{3/2}\sqrt{\det D_{S,J}}}, ϕ ( k ) = S v ∑ J ( v ) ( 1 − cos ( k ⋅ v )) , κ = ( 4 π ) 3/2 det D S , J 1 , the site transition kernel is
p t ( y , w ) = ∫ [ − π , π ] 3 e i k ⋅ ( w − y ) e − t ϕ ( k ) d k ( 2 π ) 3 . (64) p_t(y,w) = \int_{[-\pi,\pi]^3} e^{\mathrm{i}k\cdot(w-y)}e^{-t\phi(k)}\frac{\mathrm{d}k}{(2\pi)^3}. \tag*{(64)} p t ( y , w ) = ∫ [ − π , π ] 3 e i k ⋅ ( w − y ) e − tϕ ( k ) ( 2 π ) 3 d k . (64) Lemma 6.1 (Ideal return coefficient). The full site kernel satisfies
sup y , w p t ( y , w ) ≤ C ( 1 + t ) − 3 / 2 , p t ( x , x ) = t − 3 / 2 ( κ + o ( 1 ) ) . \sup_{y,w}p_t(y,w)\le C(1+t)^{-3/2}, \qquad p_t(x,x)=t^{-3/2}(\kappa+o(1)). y , w sup p t ( y , w ) ≤ C ( 1 + t ) − 3/2 , p t ( x , x ) = t − 3/2 ( κ + o ( 1 )) . The first estimate holds for all t ≥ 0 t\ge0 t ≥ 0 t → ∞ t\to\infty t → ∞ K K K 1 ≤ n ≤ 2 K 1\le n\le2K 1 ≤ n ≤ 2 K n n n
( W ^ ∞ ( d ) ) i i = ℓ − 1 p n β ( x , x ) = β − 3 / 2 ( ( κ / ℓ ) n − 3 / 2 + o K ( 1 ) ) . (65) (\widehat{W}_{\infty}(\mathbf{d}))_{ii}=\ell^{-1}p_{n\beta}(x,x)=\beta^{-3/2}\left((\kappa/\ell)n^{-3/2}+o_K(1)\right). \tag*{(65)} ( W ∞ ( d ) ) ii = ℓ − 1 p n β ( x , x ) = β − 3/2 ( ( κ / ℓ ) n − 3/2 + o K ( 1 ) ) . (65) The error is uniform in the word, cut, and initial slot. Always,
0 ≤ ( W ^ R ( d ) ) i i ≤ ( W ^ ∞ ( d ) ) i i . (66) 0\le(\widehat{W}_{R}(\mathbf{d}))_{ii}\le(\widehat{W}_{\infty}(\mathbf{d}))_{ii}. \tag*{(66)} 0 ≤ ( W R ( d ) ) ii ≤ ( W ∞ ( d ) ) ii . (66) If B R ( z ) ⊆ Λ ∘ \mathcal{B}_R(z)\subseteq\Lambda^\circ B R ( z ) ⊆ Λ ∘
0 ≤ ( W ^ ∞ ( d ) ) i i − ( W ^ R ( d ) ) i i ≤ C K e − c β . (67) 0\le(\widehat{W}_{\infty}(\mathbf{d}))_{ii}-(\widehat{W}_{R}(\mathbf{d}))_{ii}\le C_K e^{-c\beta}. \tag*{(67)} 0 ≤ ( W ∞ ( d ) ) ii − ( W R ( d ) ) ii ≤ C K e − c β . (67) Proof. At a zero of ϕ \phi ϕ k ⋅ v k\cdot v k ⋅ v J ( v ) > 0 J(v)>0 J ( v ) > 0 2 π Z 2\pi\mathbb{Z} 2 π Z v v v Z 3 \mathbb{Z}^3 Z 3 k ∈ 2 π Z 3 k\in2\pi\mathbb{Z}^3 k ∈ 2 π Z 3
ϕ ( k ) = k T D S , J k + O ( ∣ k ∣ 4 ) . \phi(k)=k^{\mathsf T}D_{S,J}k+O(|k|^4). ϕ ( k ) = k T D S , J k + O ( ∣ k ∣ 4 ) . The matrix is positive definite because the support spans R 3 \mathbb{R}^3 R 3 ϕ ( k ) ≥ c ∣ k ∣ 2 \phi(k)\ge c|k|^2 ϕ ( k ) ≥ c ∣ k ∣ 2 [ − π , π ] 3 [-\pi,\pi]^3 [ − π , π ] 3 t ≤ 1 t\le1 t ≤ 1 k = u / t k=u/\sqrt{t} k = u / t e − c ∣ u ∣ 2 e^{-c|u|^2} e − c ∣ u ∣ 2
∫ R 3 e − u T D S , J u d u ( 2 π ) 3 = 1 ( 4 π ) 3 / 2 det D S , J . \int_{\mathbb{R}^3}e^{-u^{\mathsf T}D_{S,J}u}\frac{\mathrm{d}u}{(2\pi)^3}=\frac{1}{(4\pi)^{3/2}\sqrt{\det D_{S,J}}}. ∫ R 3 e − u T D S , J u ( 2 π ) 3 d u = ( 4 π ) 3/2 det D S , J 1 . This proves (6.2).
Averaging a uniform site permutation makes the slot index uniform conditional on the site. The inverse of a uniform permutation has the same property. Every complete traversal contains one seam: if the cut is zero, it is at the end of a forward traversal and at the start of a reverse traversal. Free exchange transitions preserve this conditional uniformity, since the rate to each target slot is the same and the exit rate does not depend on the starting slot. Thus after the first seam, the slot index is uniform conditional on the site. The site process runs for total elapsed time n β n\beta n β 1 ≤ n ≤ 2 K 1\le n\le2K 1 ≤ n ≤ 2 K
Killing restricts a path event, proving (6.4). Under the interior hypothesis it cannot act before the full ideal path leaves B R ( z ) \mathcal{B}_R(z) B R ( z ) 2 K 2K 2 K C K e − c β C_K e^{-c\beta} C K e − c β
Comparison in one exposure
Proposition 6.2 (Finite direction words). For every fixed K K K
∣ E P [ ( W R ( d ) ) i i ] − ( W ^ R ( d ) ) i i ∣ = o K ( β − 3 / 2 ) . (68) \left|\mathbb{E}_P[(W_R(\mathrm d))_{ii}] - (\widehat{W}_R(\mathrm d))_{ii}\right| = o_K(\beta^{-3/2}). \tag*{(68)} E P [( W R ( d ) ) ii ] − ( W R ( d ) ) ii = o K ( β − 3/2 ) . (68) The error is uniform over 1 ≤ n ≤ 2 K 1 \le n \le2K 1 ≤ n ≤ 2 K n n n E E E
Proof. Midpoint reduction. Split the elapsed word at time n β / 2 n\beta/2 n β /2
W R = B C , W ^ R = B ^ C ^ . W_R = BC,\qquad\widehat{W}_R = \widehat{B}\widehat{C}. W R = B C , W R = B C . Here C C C η = 1 { i } \eta= 1_{\{i\}} η = 1 { i }
μ 1 = C η , μ 2 = B ∗ η , μ ^ 1 = C ^ η , μ ^ 2 = B ^ ∗ η . \mu_1 = C\eta,\qquad\mu_2 = B^*\eta,\qquad\widehat{\mu}_1 = \widehat{C}\eta,\qquad\widehat{\mu}_2 = \widehat{B}^*\eta. μ 1 = C η , μ 2 = B ∗ η , μ 1 = C η , μ 2 = B ∗ η . After extending hole vectors by zero, all four are nonnegative masses of total at most one on the same full slot layer. Consequently
( W R ) i i = ⟨ μ 1 , μ 2 ⟩ , ( W ^ R ) i i = ⟨ μ ^ 1 , μ ^ 2 ⟩ . (69) (W_R)_{ii} = \langle\mu_1,\mu_2\rangle,\qquad(\widehat{W}_R)_{ii} = \langle\widehat{\mu}_1,\widehat{\mu}_2\rangle. \tag*{(69)} ( W R ) ii = ⟨ μ 1 , μ 2 ⟩ , ( W R ) ii = ⟨ μ 1 , μ 2 ⟩ . (69) The second mass is propagated backward from the terminal copy of i i i Figure 1 .
Figure 1. A diagonal return is an inner product on the midpoint layer. Both masses use the same exposure, with fresh transitions in each traversal. Coupling either mass to an independent full ideal path gives an expected ℓ 1 \ell^1 ℓ 1 ℓ ∞ \ell^\infty ℓ ∞
Let G G G z , R z,R z , R n β / 2 ≥ β / 2 n\beta/2 \ge\beta/2 n β /2 ≥ β /2 ∥ μ 1 ∥ ∞ + ∥ μ 2 ∥ ∞ ≤ C β − 3 / 2 \lVert\mu_1\rVert_\infty+ \lVert\mu_2\rVert_\infty\le C\beta^{-3/2} ∥ μ 1 ∥ ∞ + ∥ μ 2 ∥ ∞ ≤ C β − 3/2 G G G
∣ E P [ ( W R ) i i ] − ( W ^ R ) i i ∣ ≤ C β − 3 / 2 E P [ ∥ μ 1 − μ ^ 1 ∥ 1 + ∥ μ 2 − μ ^ 2 ∥ 1 ] + P P ( G c ) . (70) \left|\mathbb{E}_P[(W_R)_{ii}] - (\widehat{W}_R)_{ii}\right|
\le C\beta^{-3/2}\mathbb{E}_P\left[\lVert\mu_1-\widehat{\mu}_1\rVert_1 + \lVert\mu_2-\widehat{\mu}_2\rVert_1\right] + \mathbb{P}_P(G^c). \tag*{(70)} E P [( W R ) ii ] − ( W R ) ii ≤ C β − 3/2 E P [ ∥ μ 1 − μ 1 ∥ 1 + ∥ μ 2 − μ 2 ∥ 1 ] + P P ( G c ) . (70) Indeed on G G G G c G^c G c [ 0 , 1 ] [0,1] [ 0 , 1 ] P P ( G c ) = o K ( β − 3 / 2 ) \mathbb{P}_P(G^c) = o_K(\beta^{-3/2}) P P ( G c ) = o K ( β − 3/2 )
E P ∥ μ j − μ ^ j ∥ 1 = o K ( 1 ) , j = 1 , 2. (71) \mathbb{E}_P\lVert\mu_j-\widehat{\mu}_j\rVert_1 = o_K(1),\qquad j=1,2. \tag*{(71)} E P ∥ μ j − μ j ∥ 1 = o K ( 1 ) , j = 1 , 2. (71) Coupling conditional on the exposure. Both masses are generated from i i i j = 2 j=2 j = 2 P P \mathbb{P}_{P} P P B P \mathcal{B}_{P} B P λ J \lambda_{J} λ J
Kill both paths outside Λ ∘ \Lambda^{\circ} Λ ∘ B R ( z ) \mathcal{B}_{R}(z) B R ( z ) Λ ∘ \Lambda^{\circ} Λ ∘ Λ \Lambda Λ 2 B ∗ 2B_{*} 2 B ∗
Averaged encounter bound. We bound the probability of this encounter unconditionally. Partition the schedule into O K ( β ) O_{K}(\beta) O K ( β ) h β = ⌈ log β ⌉ h_{\beta}=\lceil\log\beta\rceil h β = ⌈ log β ⌉ w w w B ∗ B_{*} B ∗ B ∗ h β B_{*}h_{\beta} B ∗ h β
C h β h β ! . (72) \frac{C^{h_{\beta}}}{h_{\beta}!}. \tag*{(72)} h β ! C h β . (72) For a deterministic w w w P P P
P P ( an exposed slot lies over B 4 B ∗ ( h β + 1 ) ( w ) at the initial slice ) ≤ C ( h β + 1 ) 3 β − 3 / 2 . (73) \mathbb{P}_{P}\left(\text{an exposed slot lies over }\mathcal{B}_{4B_{*}(h_{\beta}+1)}(w)\text{ at the initial slice}\right)\le C(h_{\beta}+1)^{3}\beta^{-3/2}. \tag*{(73)} P P ( an exposed slot lies over B 4 B ∗ ( h β + 1 ) ( w ) at the initial slice ) ≤ C ( h β + 1 ) 3 β − 3/2 . (73) At a seam the set of sites occupied by exposed slots is the same on both sides, so this estimate covers either endpoint convention.
If this initial event is absent but an exposed strand subsequently visits B B ∗ ( h β + 2 ) ( w ) \mathcal{B}_{B_{*}(h_{\beta+2)}(w)} B B ∗ ( h β + 2 ) ( w ) h β h_{\beta} h β h β h_{\beta} h β β \beta β β \beta β β \beta β C ( h β + 1 ) 3 C(h_{\beta}+1)^{3} C ( h β + 1 ) 3
C ( h β + 1 ) 3 C h β ∫ 0 < s 1 < ⋯ < s h β < 1 d s 1 ⋯ d s h β = C ( h β + 1 ) 3 C h β h β ! . (74) C(h_{\beta}+1)^{3}C^{h_{\beta}}\int_{0<s_{1}<\cdots<s_{h_{\beta}}<1} ds_{1}\cdots ds_{h_{\beta}}
= C(h_{\beta}+1)^{3}\frac{C^{h_{\beta}}}{h_{\beta}!}. \tag*{(74)} C ( h β + 1 ) 3 C h β ∫ 0 < s 1 < ⋯ < s h β < 1 d s 1 ⋯ d s h β = C ( h β + 1 ) 3 h β ! C h β . (74) Discarding slot compatibility only enlarges the count. A seam can change a slot index but no site, so it merely contributes a bounded slot choice; unwrapping preserves the simplex volume. Reversing a traversal reverses the order of the times and gives the same bound. Thus (6.12) uses factorial domination under the full picture law, without conditioning that domination on the exposure.
The random coordinate w w w (73) and (74) , uniform for deterministic w w w w w w
C K β ( h β + 1 ) 3 ( β − 3 / 2 + C h β h β ! ) = o K ( 1 ) . (75) C_K \beta(h_\beta+1)^3\left(\beta^{-3/2}+\frac{C h^\beta}{h_\beta!}\right)=o_K(1). \tag*{(75)} C K β ( h β + 1 ) 3 ( β − 3/2 + h β ! C h β ) = o K ( 1 ) . (75) The first term is O K ( β − 1 / 2 ( log β ) 3 ) O_K(\beta^{-1/2}(\log\beta)^3) O K ( β − 1/2 ( log β ) 3 ) β \beta β
Add a death state for killing. Conditional on each exposure, the ℓ 1 \ell^1 ℓ 1 (75) . This proves (71) for both schedules and, by (70) , proves (68) . □
Theorem 6.3 (One-slot bounds with the sharp coefficient). There is a constant C < ∞ C<\infty C < ∞ K K K K ≥ 1 K\ge1 K ≥ 1 β K < ∞ \beta_K<\infty β K < ∞ ϵ K ( β ) → 0 \epsilon_K(\beta)\to0 ϵ K ( β ) → 0 β → ∞ \beta\to\infty β → ∞ β ≥ β K \beta\ge\beta_K β ≥ β K E E E i i i
d E ( i ) ≤ β − 3 / 2 ( κ ℓ ∑ n = 1 2 K − 1 n − 3 / 2 + C K − 1 / 2 + ϵ K ( β ) ) . (76) d_E(i)\le\beta^{-3/2}\left(\frac{\kappa}{\ell}\sum_{n=1}^{2K-1}n^{-3/2}+CK^{-1/2}+\epsilon_K(\beta)\right). \tag*{(76)} d E ( i ) ≤ β − 3/2 ( ℓ κ n = 1 ∑ 2 K − 1 n − 3/2 + C K − 1/2 + ϵ K ( β ) ) . (76) If the site x x x i i i
B R K ( x ˉ ) ⊆ Λ , E has no pin over B R K ( x ˉ ) , (77) \mathcal{B}_{R_K}(\bar{x})\subseteq\Lambda,\qquad E\text{ has no pin over }\mathcal{B}_{R_K}(\bar{x}), \tag*{(77)} B R K ( x ˉ ) ⊆ Λ , E has no pin over B R K ( x ˉ ) , (77) then
d E ( i ) ≥ β − 3 / 2 ( κ ℓ ∑ n = 1 K n − 3 / 2 − ϵ K ( β ) ) . (78) d_E(i)\ge\beta^{-3/2}\left(\frac{\kappa}{\ell}\sum_{n=1}^{K}n^{-3/2}-\epsilon_K(\beta)\right). \tag*{(78)} d E ( i ) ≥ β − 3/2 ( ℓ κ n = 1 ∑ K n − 3/2 − ϵ K ( β ) ) . (78) For i ∉ E i\notin E i ∈ / E r ( E , { i } ) r(E,\{i\}) r ( E , { i }) K K K
Proof. Suppose first that i ∉ E i\notin E i ∈ / E P P \mathbb{P}_P P P P = E ∪ { i } P=E\cup\{i\} P = E ∪ { i }
r ( E , { i } ) ≤ ∑ n = 1 2 K − 1 E P ⟨ η , Q n η ⟩ + β − 3 / 2 ( C K − 1 / 2 + o K ( 1 ) ) . r(E,\{i\})\le\sum_{n=1}^{2K-1}\mathbb{E}_P\langle\eta,Q^n\eta\rangle+\beta^{-3/2}(CK^{-1/2}+o_K(1)). r ( E , { i }) ≤ n = 1 ∑ 2 K − 1 E P ⟨ η , Q n η ⟩ + β − 3/2 ( C K − 1/2 + o K ( 1 )) . Expand Q n Q^n Q n 2 n 2^n 2 n C K e − c β CKe^{-c\beta} C K e − c β
r ( E , { i } ) ≤ β − 3 / 2 ( κ ℓ ∑ n = 1 2 K − 1 n − 3 / 2 + C K − 1 / 2 + o K ( 1 ) ) . r(E,\{i\})\le\beta^{-3/2}\left(\frac{\kappa}{\ell}\sum_{n=1}^{2K-1}n^{-3/2}+CK^{-1/2}+o_K(1)\right). r ( E , { i }) ≤ β − 3/2 ( ℓ κ n = 1 ∑ 2 K − 1 n − 3/2 + C K − 1/2 + o K ( 1 ) ) . There are only finitely many words at fixed K K K o K ( 1 ) o_K(1) o K ( 1 ) K − 1 / 2 K^{-1/2} K − 1/2
For the lower bound assume (77) . At fixed exposure, run the forward fresh walk with boundary and ball killing, and continue it after visits to i i i A n A_n A n i i i n β n\beta n β E E E q i ≥ Pr ( ⋃ n = 1 K A n ∣ exposure ) q_i \ge\Pr\left(\bigcup_{n=1}^{K} A_n \mid\text{exposure}\right) q i ≥ Pr ( ⋃ n = 1 K A n ∣ exposure )
Pr ( A n ∣ exposure ) = ( U + , R n ) i i , Pr ( A j ∩ A n ∣ exposure ) = ( U + , R j ) i i ( U + , R n − j ) i i ( j < n ) . \Pr(A_n \mid\text{exposure})=(U_{+,R}^{n})_{ii}, \qquad\Pr(A_j\cap A_n \mid\text{exposure})=(U_{+,R}^{j})_{ii}(U_{+,R}^{n-j})_{ii}\quad(j<n). Pr ( A n ∣ exposure ) = ( U + , R n ) ii , Pr ( A j ∩ A n ∣ exposure ) = ( U + , R j ) ii ( U + , R n − j ) ii ( j < n ) . The Bonferroni lower bound therefore reads
q i ≥ ∑ n = 1 K ( U + , R n ) i i − ∑ 1 ≤ j < n ≤ K ( U + , R j ) i i ( U + , R n − j ) i i . (79) q_i \ge\sum_{n=1}^{K}(U_{+,R}^{n})_{ii}-\sum_{1\le j<n\le K}(U_{+,R}^{j})_{ii}(U_{+,R}^{n-j})_{ii}. \tag*{(79)} q i ≥ n = 1 ∑ K ( U + , R n ) ii − 1 ≤ j < n ≤ K ∑ ( U + , R j ) ii ( U + , R n − j ) ii . (79) On good exposures each product is O ( β − 3 ) O(\beta^{-3}) O ( β − 3 ) o K ( β − 3 / 2 ) o_K(\beta^{-3/2}) o K ( β − 3/2 ) o K ( β − 3 / 2 ) o_K(\beta^{-3/2}) o K ( β − 3/2 ) (67) . Averaging (79) gives
r ( E , { i } ) ≥ β − 3 / 2 ( κ ℓ ∑ n = 1 K n − 3 / 2 − o K ( 1 ) ) . r(E,\{i\})\ge\beta^{-3/2}\left(\frac{\kappa}{\ell}\sum_{n=1}^{K}n^{-3/2}-o_K(1)\right). r ( E , { i }) ≥ β − 3/2 ( ℓ κ n = 1 ∑ K n − 3/2 − o K ( 1 ) ) . Finally, Proposition 2.4 and Corollary 5.3 give, uniformly,
d E ( i ) = r ( E , { i } ) 1 + r ( E , { i } ) , 0 ≤ r ( E , { i } ) − d E ( i ) ≤ C β − 3 . d_E(i)=\frac{r(E,\{i\})}{1+r(E,\{i\})},\qquad0\le r(E,\{i\})-d_E(i)\le C\beta^{-3}. d E ( i ) = 1 + r ( E , { i }) r ( E , { i }) , 0 ≤ r ( E , { i }) − d E ( i ) ≤ C β − 3 . This changes only the normalized o K ( 1 ) o_K(1) o K ( 1 ) β K \beta_K β K r r r d d d i ∈ E i\in E i ∈ E d E ( i ) = 0 d_E(i)=0 d E ( i ) = 0
From pinned returns to spontaneous magnetization
The return estimates hold for every deterministic choice of extra pins. A positive magnetic field is an exact mixture of such choices. We use this fact to bound finite-volume pressure increments, then take the thermodynamic limit before differentiating at zero field.
Pressure and the cost of boundary pins
Write V = ∣ Λ N ∣ V=|\Lambda_N| V = ∣ Λ N ∣
p N ( h ) = 1 β V log Z Λ N ( h ) . p_N(h)=\frac{1}{\beta V}\log Z_{\Lambda_N}(h). p N ( h ) = β V 1 log Z Λ N ( h ) . The limit p β ( h ) = lim N → ∞ p N ( h ) p_\beta(h)=\lim_{N\to\infty}p_N(h) p β ( h ) = lim N → ∞ p N ( h ) β > 0 \beta>0 β > 0 h ∈ R h\in\mathbb{R} h ∈ R ℓ \ell ℓ O ( V / ℓ ) O(V/\ell) O ( V / ℓ ) o ( V ) o(V) o ( V ) N → ∞ N\to\infty N → ∞ ℓ \ell ℓ a a a β a \beta a β a C / ℓ C/\ell C / ℓ ℓ → ∞ \ell\to\infty ℓ → ∞ [ H S , N J , M S , N ] = 0 [H^J_{S,N},M_{S,N}]=0 [ H S , N J , M S , N ] = 0 ∥ M S , N ∥ ≤ S V \|M_{S,N}\|\le SV ∥ M S , N ∥ ≤ S V p N p_N p N S S S h h h p β p_\beta p β
Let Z Λ N + Λ ( h ) Z_{\Lambda_N}^{+\Lambda}(h) Z Λ N + Λ ( h ) D 0 D_0 D 0 + 1 +1 + 1 p N + ( h ) = ( β V ) − 1 log Z Λ N + Λ ( h ) p_N^+(h)=(\beta V)^{-1}\log Z_{\Lambda_N}^{+\Lambda}(h) p N + ( h ) = ( β V ) − 1 log Z Λ N + Λ ( h )
Proposition 7.1 (Boundary pressure equivalence). For every fixed β > 0 \beta> 0 β > 0 and h ≥ 0 h \ge0 h ≥ 0
lim N → ∞ p N + ( h ) = p β ( h ) . (80) \lim_{N \to\infty} p_N^{+}(h) = p_\beta(h). \tag*{(80)} N → ∞ lim p N + ( h ) = p β ( h ) . (80) Proof. Factor e β h S V e^{\beta h S_V} e β h S V
W h = ∏ C ( 1 + e − β h a C ) . W_h = \prod_C (1 + e^{-\beta h a_C}). W h = C ∏ ( 1 + e − β h a C ) . Given the picture, the colors of different cycles are independent and the probability that a cycle is up is at least 1 / 2 1/2 1/2 b ∂ b_{\partial} b ∂ D 0 D_0 D 0
1 ≥ Z Λ + ( h ) Z Λ ( h ) ≥ E h [ 2 − b ∂ ] ≥ exp { − log ( 2 ) E h b ∂ } . (81) 1 \ge\frac{Z_\Lambda^{+}(h)}{Z_\Lambda(h)} \ge\mathbb{E}_h[2^{-b_{\partial}}] \ge\exp\{-\log(2)\mathbb{E}_h b_{\partial}\}. \tag*{(81)} 1 ≥ Z Λ ( h ) Z Λ + ( h ) ≥ E h [ 2 − b ∂ ] ≥ exp { − log ( 2 ) E h b ∂ } . (81) A cycle visiting the boundary either contains a time-zero boundary slot or arrives there at a mark incident to the boundary. Thus
b ∂ ≤ ℓ ∣ ∂ Λ ∣ + 2 N ∂ , b_{\partial} \le\ell|\partial\Lambda| + 2N_{\partial}, b ∂ ≤ ℓ ∣ ∂ Λ∣ + 2 N ∂ , where N ∂ N_{\partial} N ∂ W h W_h W h W h W_h W h E h N ∂ ≤ C β ∣ ∂ Λ ∣ \mathbb{E}_hN_{\partial} \le C\beta|\partial\Lambda| E h N ∂ ≤ C β ∣ ∂ Λ∣
0 ≤ p N ( h ) − p N + ( h ) ≤ C ( 1 + β ) ∣ ∂ Λ ∣ β ∣ Λ ∣ . 0 \le p_N(h) - p_N^{+}(h) \le C(1+\beta)\frac{|\partial\Lambda|}{\beta|\Lambda|}. 0 ≤ p N ( h ) − p N + ( h ) ≤ C ( 1 + β ) β ∣Λ∣ ∣ ∂ Λ∣ . For fixed β \beta β
A magnetic field as random extra pins
The mixture in the next lemma is useful because its conditional laws are precisely the deterministic pin laws already studied. Its mixing measure need not be a product measure.
Lemma 7.2 (Extra-pin mixture). For h ≥ 0 h \ge0 h ≥ 0 the colored-picture law defining Z Λ + ( h ) Z_\Lambda^{+}(h) Z Λ + ( h ) is a mixture of the zero-field pinned laws with pin sets E = D 0 ∪ F E = D_0 \cup F E = D 0 ∪ F where F F F consists of time-zero slots. Under the mixing law,
Pr ( i ∈ F ) ≤ b ( h ) : = 1 − e − β h for every time-zero slot i . \Pr(i \in F) \le b(h) := 1 - e^{-\beta h}
\qquad\text{for every time-zero slot } i. Pr ( i ∈ F ) ≤ b ( h ) := 1 − e − β h for every time-zero slot i . Moreover,
S − ( p N + ) ′ ( h ) = 1 V ∑ i ∈ V Λ E F [ d D 0 ∪ F ( i ) ] . (82) S - (p_N^{+})'(h) = \frac{1}{V}\sum_{i \in V_\Lambda}\mathbb{E}_F[d_{D_0\cup F}(i)]. \tag*{(82)} S − ( p N + ) ′ ( h ) = V 1 i ∈ V Λ ∑ E F [ d D 0 ∪ F ( i )] . (82) Proof. For a colored picture let D d o w n D_{\mathrm{down}} D down M = S V − D d o w n M = SV - D_{\mathrm{down}} M = S V − D down b ( h ) b(h) b ( h ) ( 1 − b ( h ) ) D d o w n = e − β h D d o w n (1-b(h))^{D_{\mathrm{down}}} = e^{-\beta hD_{\mathrm{down}}} ( 1 − b ( h ) ) D down = e − β h D down 1 F \mathbf{1}_F 1 F F F F D 0 ∪ F D_0 \cup F D 0 ∪ F 2 n D 0 ∪ F 2^{n_{D_0\cup F}} 2 n D 0 ∪ F
Conditional on the picture and its colors in this joint law, down slots are never selected, while up slots are selected independently with probability b ( h ) b(h) b ( h ) ( p N + ) ′ ( h ) = V − 1 E M (p_N^{+})'(h) = V^{-1}\mathbb{E}M ( p N + ) ′ ( h ) = V − 1 E M F F F
The order of limits
Recall the ideal heat-kernel constant and define its finite sums by
κ = 1 ( 4 π ) 3 / 2 det D S , J , c j = κ ∑ n = 1 j n − 3 / 2 . (83) \kappa= \frac{1}{(4\pi)^{3/2}\sqrt{\det D_{S,J}}}, \qquad c_j = \kappa\sum_{n=1}^{j} n^{-3/2}. \tag*{(83)} κ = ( 4 π ) 3/2 det D S , J 1 , c j = κ n = 1 ∑ j n − 3/2 . (83) Fix an integer K ≥ 1 K \ge1 K ≥ 1 R = R K = ⌈ a ( K + 1 ) β ⌉ R = R_K = \lceil a(K+1)\beta\rceil R = R K = ⌈ a ( K + 1 ) β ⌉
S − ( p N + ) ′ ( h ) ≤ β − 3 / 2 ( c 2 K − 1 + C K − 1 / 2 + ϵ K ( β ) ) . (84) S - (p_N^+)'(h) \le\beta^{-3/2}\left(c_{2K-1}+CK^{-1/2}+\epsilon_K(\beta)\right). \tag*{(84)} S − ( p N + ) ′ ( h ) ≤ β − 3/2 ( c 2 K − 1 + C K − 1/2 + ϵ K ( β ) ) . (84) where ϵ K ( β ) ≥ 0 \epsilon_K(\beta) \ge0 ϵ K ( β ) ≥ 0 β → ∞ \beta\to\infty β → ∞ K K K N N N h ≥ 0 h \ge0 h ≥ 0 C C C K K K ℓ V \ell V ℓ V 1 / ℓ 1/\ell 1/ ℓ
For a lower bound, let ρ N \rho_N ρ N x ∈ Λ N x \in\Lambda_N x ∈ Λ N B R ( x ˉ ) ⊆ Λ N ∘ \mathcal{B}_R(\bar{x}) \subseteq\Lambda_N^\circ B R ( x ˉ ) ⊆ Λ N ∘ A A A x x x R R R ρ N → 1 \rho_N \to1 ρ N → 1 N → ∞ N \to\infty N → ∞ β \beta β K K K
v R = ℓ ∣ E ∣ ( 2 R + 1 ) 3 v_R = \ell|{\cal E}|(2R+1)^3 v R = ℓ ∣ E ∣ ( 2 R + 1 ) 3 time-zero slots above the ball. The union bound and (7.3) show that it contains no extra pin with probability at least [ 1 − v R b ( h ) ] + [1-v_Rb(h)]_+ [ 1 − v R b ( h ) ] + [ u ] + = max { u , 0 } [u]_+ = \max\{u,0\} [ u ] + = max { u , 0 } ϵ K \epsilon_K ϵ K
S − ( p N + ) ′ ( h ) ≥ ρ N [ 1 − v R b ( h ) ] + β − 3 / 2 ( c K − ϵ K ( β ) ) . S - (p_N^+)'(h) \ge\rho_N[1-v_Rb(h)]_+\beta^{-3/2}(c_K-\epsilon_K(\beta)). S − ( p N + ) ′ ( h ) ≥ ρ N [ 1 − v R b ( h ) ] + β − 3/2 ( c K − ϵ K ( β )) . Here and below β \beta β K K K c K − ϵ K ( β ) ≥ 0 c_K-\epsilon_K(\beta) \ge0 c K − ϵ K ( β ) ≥ 0
Proof of Theorem 1.1. Integrate (7.6) and (7.7) from 0 0 0 h > 0 h>0 h > 0 h h h
S − p N + ( h ) − p N + ( 0 ) h . S-\frac{p_N^+(h)-p_N^+(0)}{h}. S − h p N + ( h ) − p N + ( 0 ) . First send N → ∞ N \to\infty N → ∞ ρ N → 1 \rho_N \to1 ρ N → 1 h ↓ 0 h \downarrow0 h ↓ 0 β \beta β K K K b b b
lim h ↓ 0 1 h ∫ 0 h [ 1 − v R b ( u ) ] + d u = 1. \lim_{h\downarrow0}\frac{1}{h}\int_0^h[1-v_Rb(u)]_+\,du=1. h ↓ 0 lim h 1 ∫ 0 h [ 1 − v R b ( u ) ] + d u = 1. By the definition of the right derivative,
c K − ϵ K ( β ) ≤ β 3 / 2 ( S − m S , J ( β ) ) ≤ c 2 K − 1 + C K − 1 / 2 + ϵ K ( β ) . c_K-\epsilon_K(\beta) \le\beta^{3/2}(S-m_{S,J}(\beta)) \le c_{2K-1}+CK^{-1/2}+\epsilon_K(\beta). c K − ϵ K ( β ) ≤ β 3/2 ( S − m S , J ( β )) ≤ c 2 K − 1 + C K − 1/2 + ϵ K ( β ) . Thus the thermodynamic limit and the zero-field derivative have been handled through pressure increments. Now let β → ∞ \beta\to\infty β → ∞ K K K K → ∞ K \to\infty K → ∞ κ ζ ( 3 / 2 ) \kappa\zeta(3/2) κ ζ ( 3/2 ) □ \square □
References 
