OAI.SKGap.sk_main

The theorem states that, for every inverse-temperature parameter β with 0<β<1, the Sherrington–Kirkpatrick high-temperature statement MainStatement(β) holds. Here a spin configuration on n sites is a Boolean assignment (spin values +1 or −1), and the disorder is a real coupling g on each unordered pair i<k, extended symmetrically with zero diagonal. With zero external field, the Hamiltonian is H(x)=½∑ᵢ∑ₖ σᵢ gᵢₖ σₖ, the Gibbs weight is exp(H(x)), and the Gibbs measure is the weight divided by the partition function. Variance is taken under this measure, and the Dirichlet form is the sum over sites i of the Gibbs expectation of (f(x) − E[f | all spins other than i])², where the conditional expectation averages f over the two values of spin i with Gibbs weights. The discrete gap is the infimum, over functions f of positive variance, of Dirichlet(f)/(n·Var(f)). The disorder law makes the edge couplings independent centered Gaussians with variance β²/n. MainStatement(β) asserts that there exists a constant C>0 such that, as n→∞, both the probability that the Poincaré inequality Var(f) ≤ C·Dirichlet(f) holds for all f: {−1,1}ⁿ→ℝ, and the probability that the discrete gap is at least 1/(C·n), tend to 1.