Propagation of errors along microscopic paths

Then the microscopic flow through an admissible bound, b = σ ℓ , σ ℓ+1 , is equal to f x σ ℓ , σ ℓ+1 = P x b ∈ Σ = ν x ℓ σ ℓ q ℓ σ ℓ , σ ℓ+1 = ν x ℓ σ ℓ Λ − k ℓ σ ℓ 1 + Oε . 5.62 Consequently, the expression in 5.59, and hence the second term in 5.56, is equal to ν x ℓ σ ℓ µ x ℓ β,N σ ℓ 1 + Oε ≡ Ψ ℓ σ ℓ 1 + Oε . 5.63 Main result. We claim that there exists a set, T A ,B , of good mesoscopic trajectories from A to B, such that P f A ,B N € X A ,B ∈ T A ,B Š = 1 − o 1, 5.64 and, uniformly in x ∈ T A ,B , E x    ℓ B −1 X ℓ=−ℓ A Ψ ℓ σ ℓ φ A ,B x ℓ , x ℓ+1    ≤ 1 + Oε. 5.65 This will imply that, capA, B ≥ Φ N eg 1 − Oε , 5.66 which is the lower bound necessary to prove Theorem 1.3. The rest of the Section is devoted to the proof of 5.65. First of all we derive recursive estimates on Ψ ℓ for a given realization, x , of the mesoscopic chain. After that it will be obvious how to define T A ,B .

5.6 Propagation of errors along microscopic paths

Let x be given. Notice that µ x ℓ β,N is the product measure, µ x ℓ β,N = n O j=1 µ x ℓ j β,N , 5.67 where µ x ℓ j β,N is the corresponding canonical measure on the mesoscopic slot S j N = {−1, 1} Λ j . On the other hand, according to 5.61, the big microscopic chain Σ splits into a direct product of n small microscopic chains, Σ 1 , . . . , Σ n , which independently evolve on S 1 N , . . . , S n N . Thus, k ℓ = k means that the ℓ-th step of the mesoscopic chain induces a step of the k-th small microscopic chain Σ k . Let τ 1 [ℓ], . . . , τ n [ℓ] be the numbers of steps performed by each of the small microscopic chains after ℓ steps of the mesoscopic chain or, equivalently, after ℓ steps of the big microscopic chain Σ. Then the corrector, Ψ ℓ , in 5.63 equals Ψ ℓ σ ℓ = n Y j=1 ψ j τ j [ℓ] σ j ℓ , 5.68 1580 where σ j ℓ is the projection of σ ℓ on S j N . Therefore we are left with two separate tasks: On the microscopic level we need to control the propagation of errors along small chains and, on the mesoscopic level, we need to control the statistics of τ 1 [ℓ], . . . , τ n [ℓ]. The latter task is related to characterizing the set, T A ,B , of good mesoscopic trajectories and it is relegated to Subsection 5.7 Small microscopic chains. It would be convenient to study the propagation of errors along small microscopic chains in the following slightly more general context: Fix 1 ≪ M ∈ N and 0 ≤ ε ≪ 1. Let g 1 , . . . , g M ∈ [−1, 1]. Consider spin configurations, ξ ∈ S M = {−1, 1} M , with product weights w ξ = e ε P i g i ξi . 5.69 As before, let Λ ± ξ = {i : ξi = ±1}. Define layers of fixed magnetization, S M [K] = ¦ ξ ∈ S M : Λ + ξ = K © . Finally, fix δ , δ 1 ∈ 0, 1, such that δ δ 1 . Set K = ⌊δ M ⌋ and r = ⌊δ 1 − δ M ⌋. We consider a Markov chain, Ξ = Ξ , . . . , Ξ r on S M , such that Ξ τ ∈ S M [K + τ] ≡ S τ M for τ = 0, 1, . . . , r. Let µ τ be the canonical measure, µ τ ξ = w ξ 1 { ξ∈S τ M } Z τ . 5.70 We take ν = µ as the initial distribution of Ξ and, following 5.61, we define transition rates, q τ ξ τ , θ + j ξ τ = e 2 εg j P i ∈Λ − ξ τ e 2 εg i . 5.71 We denote by P the law of this Markov chain and let ν τ be the distribution of Ξ τ which is concen- trated on S τ M , that is, ν τ ξ = P Ξ τ = ξ . The propagation of errors along paths of our chain is then quantified in terms of ψ τ · ≡ ν τ ·µ τ ·. Proposition 5.1. For every τ = 1, . . . , r and each ξ ∈ S τ M define B τ ξ ≡ M X i=1 e 2 εg i 1 { i ∈Λ − ξ } and A τ = µ τ B τ · = M X i=1 e 2 εg i µ τ € i ∈ Λ − · Š . 5.72 Then there exists c = c δ , δ 1 such that the following holds: For any trajectory, ξ = ξ , . . . , ξ r , of positive probability under P, it holds that ψ τ ξ τ ≤ A B ξ τ e c ετ 2 M , 5.73 for all τ = 0, 1, . . . , r. Proof. By construction, ψ ≡ 1. Let ξ τ+1 ∈ S τ+1 M . Since ν τ satisfies the recursion ν τ+1 ξ τ+1 = X j ∈Λ + ξ τ+1 ν τ θ − j ξ τ+1 q τ θ − j ξ τ+1 , ξ τ+1 , 5.74 1581 it follows that ψ τ satisfies ψ τ+1 ξ τ+1 = X j ∈Λ + ξ τ+1 ν τ θ − j ξ τ+1 q τ θ − j ξ τ+1 , ξ τ+1 µ τ+1 ξ τ+1 = X j ∈Λ + ξ τ+1 µ τ θ − j ξ τ+1 q τ θ − j ξ τ+1 , ξ τ+1 µ τ+1 ξ τ+1 ψ τ θ − j ξ τ+1 . By our choice of transition probabilities in 5.71, µ τ θ − j ξ τ+1 q τ θ − j ξ τ+1 , ξ τ+1 µ τ+1 ξ τ+1 = Z τ+1 Z τ    X i ∈Λ − θ − j ξ τ+1 e 2 εg i    −1 . 5.75 Recalling that Λ + ξ τ ≡ Λ + τ = K + τ does not depend on the particular value of ξ τ , Z τ+1 Z τ = 1 Z τ X ξ∈S τ+1 M w ξ = 1 Z τ X ξ∈S τ+1 M 1 Λ + ξ X j ∈Λ + ξ w θ − j ξe 2 εg j = 1 Z τ X ξ∈S τ M w ξ · 1 Λ + τ+1 X j ∈Λ − ξ e 2 εg j = µ τ    1 Λ + ξ τ+1 X j ∈Λ − · e 2 εg j    . We conclude that the right hand side of 5.75 equals 1 Λ + ξ τ+1 · µ τ P i ∈Λ − · e 2 εg i P i ∈Λ − θ − j ξ τ+1 e 2 εg i = 1 Λ + ξ τ+1 · A τ B τ θ − j ξ τ+1 . 5.76 As a result, ψ τ+1 ξ τ+1 = 1 Λ + ξ τ+1 X j ∈Λ + ξ τ+1 A τ B τ θ − j ξ τ+1 ψ τ θ − j ξ τ+1 . 5.77 Iterating the above procedure we arrive to the following conclusion: Consider the set, Dξ τ+1 , of all paths, ξ = ξ , . . . , ξ τ , ξ τ+1 , of positive probability from S M to S τ+1 M to ξ τ+1 . The number, D τ+1 ≡ Dξ τ+1 , of such paths does not depend on ξ τ+1 . Then, since ψ ≡ 1, ψ τ+1 ξ τ+1 = 1 D τ+1 X ξ∈Dξ τ+1 τ Y s=0 A s B s ξ s . 5.78 We claim that A s B s ξ s = 1 + O ε M A s −1 B s −1 ξ s −1 , 5.79 uniformly in all the quantities under consideration. Once 5.79 is verified, ψ τ ξ τ ≤ e O ετ 2 M max ξ ∼ξ τ A B ξ τ , 5.80 1582 where for ξ ∈ S M , the relation ξ ∼ ξ τ means that there is a path of positive probability from ξ to ξ τ . But all such ξ ’s differ at most in 2 τ coordinates. It is then straightforward to see that if ξ ∼ ξ τ and ξ ′ ∼ ξ τ , then B ξ B ξ ′ ≤ e O ετM , 5.81 and 5.73 follows. It remains to prove 5.79. Let ξ ∈ S s M and ξ ′ = θ − j ξ ∈ S s −1 M . Notice, first of all, that B s −1 ξ ′ − B s ξ = e 2 εg j = 1 + Oε. 5.82 Similarly, A s −1 − A s = M X i=1 e 2 εg i ¦ µ s −1 i ∈ Λ − − µ s i ∈ Λ − © = 1 + M X i=1 € e 2 εg i − 1 Š ¦ µ s −1 i ∈ Λ − − µ s i ∈ Λ − © . By usual local limit results for independent Bernoulli variables, µ s −1 i ∈ Λ − − µ s i ∈ Λ − = O 1 M , 5.83 uniformly in s = 1, . . . , r − 1 and i = 1, . . . , M. Hence, A s −1 − A s = 1 + Oε. Finally, both A s −1 and B s −1 ξ ′ are uniformly OM , whereas, A s −1 − B s −1 ξ ′ = M X i=1 € e 2 εg i − 1 Š n µ s −1 i ∈ Λ − − 1 { i ∈Λ − ξ ′ } o = OεM . 5.84 Hence, A s B s ξ = A s −1 − 1 + Oε B s −1 ξ ′ − 1 + Oε = A s −1 B s −1 ξ ′ 1 + O ε M , 5.85 which is 5.79. Back to the big microscopic chain. Going back to 5.68 we infer that the corrector of the big chain Σ satisfies the following upper bound: Let σ = σ , σ 1 , . . . be a trajectory of Σ as sampled from P x . Then, for every ℓ = 0, 1, . . . , ℓ B − 1, Ψ ℓ σ ℓ ≤ exp    c ε n X j=1 τ j [ℓ] 2 M j    n Y j=1   A j B j σ j   τ j [ℓ] , 5.86 where M j = Λ j = ρ j N , A j = X i ∈Λ j e 2˜h i µ x j β,N i ∈ Λ − j , and B j σ j = X i ∈Λ j e 2˜h i 1 n i ∈Λ − j σ j o . 5.87 1583 Of course, A j = µ x j β,N B j . It is enough to control the first order approximation,   A j B j σ j   τ j [ℓ] ≈ exp −τ j [ℓ] B j σ j − A j B j σ j ≡ exp € τ j [ℓ]Y j Š . 5.88 The variables Y 1 , . . . , Y n are independent once x is fixed. Thus, in view of our target, 5.65, we need to derive an upper bound of order 1 + O ε for E x ℓ B −1 X ℓ=0 exp    c ε n X j=1 τ j [ℓ] 2 M j + n X j=1 τ j [ℓ]Y j    φ A ,B x ℓ , x ℓ+1 = ℓ B −1 X ℓ=0 exp    c ε n X j=1 τ j [ℓ] 2 M j    n Y 1 µ x j β,N € e τ j [ℓ]Y j Š φ A ,B x ℓ , x ℓ+1 , 5.89 which holds with P f A ,B N -probability of order 1 − Oε.

5.7 Good mesoscopic trajectories