Approximated harmonic functions for e

For that concerns the rates, we first define, for σ ∈ S N , Λ ± k σ ≡ i ∈ Λ k : σi = ±1 . 4.13 For all x ∈ Γ n N , we then have r N x , x + e ℓ = Q β,N x −1 X σ∈S N [x ] µ β,N [ω]σ X i ∈Λ − ℓ σ p σ, σ i 4.14 = Q β,N x −1 X σ∈S N [x ] µ β,N [ω]σ X i ∈Λ − ℓ σ 1 N e −2β • m σ− 1 N +h i ˜ + . Notice that for all σ ∈ S N x , |Λ − ℓ σ| is a constant just depending on x . Using that h i = ¯h ℓ + e h i , with e h i ∈ [−ǫ, ǫ], we get the bounds r N x , x + e ℓ = | Λ − ℓ x | N e −2β [ m σ+¯h ℓ ] + 1 + Oǫ. 4.15 It follows easily that, for all x ∈ D N ρ, r N x , x + e ℓ r N z ∗ , z ∗ + e ℓ − 1 ≤ cβǫ + nρ, 4.16 for some finite constant c 0. With this in mind, we let erx , x + e ℓ ≡ r N z ∗ , z ∗ + e ℓ ≡ r ℓ and erx + e ℓ , x ≡ r ℓ e Q β,N x e Q β,N x +e ℓ be the modified rates of a dynamics on D N ρ reversible w.r.t. the measure e Q β,N x , and let e L N denote the correspondent generator. For u ∈ G A ,B , we write the corresponding Dirichlet form as e Φ D N u ≡ Q β,N z ∗ X x ∈D N ρ n X ℓ=1 r ℓ e −β Nx −z ∗ ,Az ∗ x −z ∗ ux − ux + e ℓ 2 . 4.17

4.3 Approximated harmonic functions for e

Φ D N We will now describe a function that we will show to be almost harmonic with respect to the Dirichlet form e Φ D N . Define the matrix Bz ∗ ≡ B with elements B ℓ,k ≡ p r ℓ A z ∗ ℓ,k p r k . 4.18 Let ˆ v i , i = 1, . . . , n be the normalized eigenvectors of B, and ˆ γ i be the corresponding eigenvalues. We denote by ˆ γ 1 the unique negative eigenvalue of B, and characterize it in the following lemma. Lemma 4.2. Let z ∗ be a solution of the equation 3.21 and assume in addition that β N N X i=1 € 1 − tanh 2 β z ∗ + h i Š 1. 4.19 1560 Then, z ∗ defined through 3.20 is a saddle point and the unique negative eigenvalue of Bz ∗ is the unique negative solution, ˆ γ 1 ≡ ˆγ 1 N , n, of the equation n X ℓ=1 ρ ℓ 1 |Λ ℓ | P i ∈Λ ℓ 1 − tanhβz ∗ + h i exp −2β ” z ∗ + ¯h ℓ — + 1 |Λℓ| P i ∈Λℓ 1 −tanhβz ∗ +h i exp −2β [ z ∗ +¯h ℓ ] + β |Λℓ| P i ∈Λℓ 1 −tanh 2 βz ∗ +h i − 2γ = 1. 4.20 Moreover, we have that lim n ↑∞ lim N ↑∞ ˆ γ 1 N , n ≡ ¯γ 1 , 4.21 where ¯ γ 1 is the unique negative solution of the equation E h     1 − tanhβz ∗ + h exp −2β [z ∗ + h] + exp −2β[z ∗ +h] + β 1+tanh βz ∗ +h − 2 γ     = 1. 4.22 Proof. The particular form of the matrix B allows to obtain a simple characterization of all eigen- values and eigenvectors. Explicitly, any eigenvector u = u 1 , . . . , u n of B with eigenvalue γ, should satisfy the set of equations − n X ℓ=1 p r ℓ r k u ℓ + r k ˆ λ k − γu k = 0, ∀ k = 1, . . . , n. 4.23 Assume for simplicity that all r k ˆ λ k take distinct values see [6], Lemma 7.2 for the general case. Then the above set of equations has no non-trivial solution for γ = r k ˆ λ k , and we can assume that P n ℓ=1 p r ℓ u ℓ 6= 0. Thus, u k = p r k P n ℓ=1 p r ℓ u ℓ r k ˆ λ k − γ . 4.24 Multiplying by pr k and summing over k, u k is a solution if and only if γ satisfies the equation n X k=1 r k r k ˆ λ k − γ = 1. 4.25 Using 4.15 and noticing that |Λ − k | N = 1 2 ρ k − z ∗ k , we get r k = 1 2 ρ k − z ∗ k exp −2β ” m σ + ¯h k — + 1 + Oǫ. 4.26 Inserting the expressions for z ∗ k ρ k and ˆ λ k given by 3.20 and 3.22 into 4.26 and substituting the result into 4.25, we recover 4.20. Since the left-hand side of 4.25 is monotone decreasing in γ as long as γ ≥ 0, it follows that there can be at most one negative solution of this equation, and such a solution exists if and only if left-hand side is larger than 1 for γ = 0. The claimed convergence property 4.21 follows easily. 1561 We continue our construction defining the vectors v i by v i ℓ ≡ ˆv i ℓ pr ℓ , 4.27 and the vectors ˇ v i by ˇ v i ℓ ≡ ˆv i ℓ p r ℓ = r ℓ v i ℓ . 4.28 We will single out the vectors v ≡ v 1 and ˇ v ≡ ˇv 1 . The important facts about these vectors is that A ˇ v i = ˆ γ i v i , 4.29 and that ˇ v i , v j = δ i j . 4.30 This implies the following non-orthogonal decomposition of the quadratic form A,

y, Ax =