Lower bound on capA, B via microscopic flows eb=e eb=e

and, in view of 3.34, we may also assume that x A ℓ x ℓ ∀x ∈ ∂ A G N and ℓ = 1, . . . , n. 5.45 Therefore, x − x A has strictly positive entries and, as it now follows from 4.29, € A ˇ v , x − x A Š = € v , x − x A Š 0. By construction G N is a small tube in the direction of ˇ v . Accordingly, we may assume that € Ax , x − x A Š 0 uniformly on ∂ A G N . But this means that the function t : [0, 1] 7→ € A x A + tx − x A , x A + tx − x A Š is strictly increasing. Therefore, F β,N is, up to negligible corrections, increasing on the straight line segment, [x A , x ] ⊂ R n which connects x A and x . Then, our target path η x is a nearest neighbor Γ n N -approximation of [x A , x ] which runs from x A to x . In view of the preceding discussion it is possible to prepare η x in such a way that F β,N z ∗ − F β,N · cN 2 δ−1 along η x . Moreover, by 5.45 it is possible to ensure that the total magnetization is increasing along η x . This concludes the construction of a flow f A ,B satisfying 5.3. In the sequel we shall index vertices of γ x = ˆ γ ∪ η x as, γ x = ˆ γ x −k A , . . . ˆ γ x . 5.46 Since, F β,N y ≤ F β,N z ∗ − c 1 y − z ∗ , v 2 , 5.47 for every y lying on the minimal energy curve ˆ x [ δ−1, z ∗ ] and since the Hessian of F β,N is uniformly bounded on ˆ x [ δ − 1, z ∗ ], we conclude that if ν is chosen small enough, then there exists c 2 such that F β,N γ x · ≤ F β,N z ∗ − c 2 γ x · − z ∗ , v 2 , 5.48 uniformly in x ∈ ∂ A G N . Finally, since the entries of v are uniformly strictly positive, it follows from 5.48 that, F β,N γ x −k ≤ F β,N z ∗ − c 3 N 1 2+δ + k 2 N 2 , 5.49 uniformly in x ∈ ∂ A and k ∈ 0, . . . , k A x .

5.5 Lower bound on capA, B via microscopic flows

Recall that A and B are mesoscopic neighborhoods of two minima of F β,N , z ∗ is the corresponding saddle point, and A = S N [A], B = S N [B] are the microscopic counterparts of A and B. Let f A ,B = f A , f, f B be the mesoscopic flow from A to B constructed above. In this section we are going to construct a subordinate microscopic flow, f

A,B

, from A to B. In the sequel, given a microscopic bond, b = σ, σ ′ , we use eb = mσ, mσ ′ for its mesoscopic pre-image. Our subordinate flow will satisfy f A ,B e = X

b:eb=e

f

A,B

b. 5.50 1577 In fact, we are going to employ a much more stringent notion of subordination on the level of induced Markov chains: Let us label the realizations of the mesoscopic chain X A ,B as x = € x −ℓ A , . . . , x ℓ B Š , in such a way that x −ℓ A ∈ A, x ℓ B ∈ B, and mx = mz ∗ . If e is a mesoscopic bond, we write e ∈ x if e = x ℓ , x ℓ+1 for some ℓ = −ℓ A , . . . , ℓ B − 1. To each path, x , of positive probability, we associate a subordinate microscopic unit flow, f x , such that f x b 0 if and only if eb ∈ x . 5.51 Then the total microscopic flow, f

A,B

, can be decomposed as f

A,B

= X x P f A ,B N € X A ,B = x Š f x . 5.52 Evidently, 5.50 is satisfied: By construction, X

b:eb=e

f x b = 1 for every x and each e ∈ x . 5.53 On the other hand, f A ,B e = P x P f A ,B N € X A ,B = x Š 1 {e∈x } . Therefore, 5.52 gives rise to the following decomposition of unity, 1 { f

A,B

b0} = X x ∋eb X σ ∋b P f A ,B N € X A ,B = x Š P x Σ = σ f A ,B eb f x b , 5.54 where P x , Σ is the microscopic Markov chain from A to B which is associated to the flow f x . Consequently, our general lower bound 2.24 implies that capA, B ≥ X x P f A ,B N € X A ,B = x Š E x    ℓ B −1 X ℓ=−ℓ A f A ,B x ℓ , x ℓ+1 f x σ ℓ , σ ℓ+1 µ β,N σ ℓ p N σ ℓ , σ ℓ+1    −1 ≥ X x P f A ,B N € X A ,B = x Š    E x ℓ B −1 X ℓ=−ℓ A f A ,B x ℓ , x ℓ+1 f x σ ℓ , σ ℓ+1 µ β,N σ ℓ p N σ ℓ , σ ℓ+1    −1 5.55 We need to recover Φ N eg from the latter expression. In view of 5.1, write, f A ,B x ℓ , x ℓ+1 f x σ ℓ , σ ℓ+1 µ β,N σ ℓ p N σ ℓ , σ ℓ+1 = φ A ,B x ℓ , x ℓ+1 Φ N eg 5.56 × Q β,N x ℓ r N x ℓ , x ℓ+1 f x σ ℓ , σ ℓ+1 µ β,N σ ℓ p N σ ℓ , σ ℓ+1 . Since we prove lower bounds, we may restrict attention to a subset of good realizations x of the mesoscopic chain X A ,B whose P f A ,B N -probability is close to one. In particular, 5.4 and 5.5 insure that the first term in the above product is precisely what we need. The remaining effort, therefore, is to find a judicious choice of f x such that the second factor in 5.56 is close to one. To this end we 1578 need some additional notation: Given a mesoscopic trajectory x = x −ℓ A , . . . , x ℓ B , define k = kℓ as the direction of the increment of ℓ-th jump. That is, x ℓ+1 = x ℓ +e k . On the microscopic level such a transition corresponds to a flip of a spin from the Λ k slot. Thus, recalling the notation Λ ± k σ ≡ i ∈ Λ k : σi = ±1 , we have that, if σ ℓ ∈ S N [x ℓ ] and σ ℓ+1 ∈ S N [x ℓ+1 ], then σ ℓ+1 = θ + i σ ℓ for some i ∈ Λ − k ℓ σ ℓ . By our choice of transition probabilities, p N , and their mesoscopic counterparts, r N , in 4.2, r N x ℓ , x ℓ+1 p N σ ℓ , σ ℓ+1 = Λ − k ℓ σ ℓ 1 + Oε , 5.57 uniformly in ℓ and in all pairs of neighbors σ ℓ , σ ℓ+1 . Note that the cardinality, Λ − k ℓ σ ℓ , is the same for all σ ℓ ∈ S N [x ℓ ]. For x ∈ Γ n N , define the canonical measure, µ x β,N σ = 1 { σ∈S N [x ] }µ β,N σ Q β,N x . 5.58 The second term in 5.56 is equal to f x σ ℓ , σ ℓ+1 µ x ℓ β,N σ ℓ · 1 Λ − k ℓ σ ℓ 1 + Oε . 5.59 If the magnetic fields, h, were constant on each set I k , then we could chose the flow f x σ ℓ , σ ℓ+1 = µ x ℓ β,N σ ℓ · 1 Λ − k ℓ σ ℓ , and consequently we would be done. In the general case of continuous distribution of h, this is not the case. However, since the fluctuations of h are bounded by 1 n, we can hope to construct f x in such a way that the ratio in 5.59 is kept very close to one. Construction of f x . We construct now a Markov chain, P x , on microscopic trajectories, Σ = ¦ σ , . . . , σ ℓ B © , from S [x ] to B, such that σ ℓ ∈ S [x ℓ ], for all ℓ = 0, . . . , ℓ B . The microscopic flow, f x , is then defined through the identity P x b ∈ Σ = f x b. The construction of a microscopic flow from A to S [x ] is completely similar it is just the reversal of the above and we will omit it. We now construct P x . S TEP 1. Marginal distributions: For each ℓ = 0, . . . , ℓ B we use ν x ℓ to denote the marginal distribution of σ ℓ under P x . The measures ν x ℓ are concentrated on S [x ℓ ]. The initial measure, ν x , is just the canonical measure µ x β,N . The measures ν x ℓ+1 are then defined through the recursive equations ν x ℓ+1 σ ℓ+1 = X σ ℓ ∈S [x ℓ ] ν x ℓ σq ℓ σ ℓ , σ ℓ+1 . 5.60 S TEP 2. Transition probabilities. The transition probabilities, q ℓ σ ℓ , σ ℓ+1 , in 5.60 are defined in the following way: As we have already remarked, all the microscopic jumps are of the form σ ℓ 7→ θ + j σ ℓ , for some j ∈ Λ − k ℓ σ, where θ + j flips the j-th spin from −1 to 1. For such a flip define q ℓ σ ℓ , θ + j σ ℓ = e 2 β˜h j P i ∈Λ − k σ ℓ e 2 β˜h i . 5.61 1579 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