OAI.BufferedIsing.finite_graph_comparison

The theorem states a comparison bound for a ferromagnetic Ising model on a finite graph with n vertices and m separately indexed edges (parallel edges allowed, no loops, nonnegative couplings K_e) and an arbitrary real external field h. A configuration σ has energy equal to the sum over edges of K_e σ_u σ_v plus the sum over vertices of h_v σ_v, with spins ±1, and weight exp(energy). Take pairwise disjoint vertex sets I, J, F, and kept vertices i ∈ I and j ∈ J that lie outside F. Fix spin configurations a and τ. For a configuration b, the conditional probability of σ_I = a given σ_J = b and σ_F = τ is the ratio of the total weight of configurations satisfying all three agreements to the total weight of those agreeing with b on J and τ on F. For any two probability mixtures mix and mix' on configurations (nonnegative weights summing to 1), the mixture probability is the mix-weighted average of these conditional probabilities over b; only the restriction of b to J matters. Let q be the terminal connection probability in the zero-field random-cluster (FK) model with q=2 on the graph with F deleted (not pinned): edges e are open with probability 1-exp(-2K_e), the vertices of I are identified together and those of J are identified together, and q is the probability that i and j lie in the same cluster. Then the ratio of the mixture probability under mix to that under mix' lies between exp(-4 artanh q) and exp(4 artanh q), for every such pair of mixtures, uniformly in h, a and τ. The theorem is stated with its proof admitted (sorry).