OAI.SK.main

The theorem states that, for every real inverse temperature β>1, the bad mass of the Sherrington-Kirkpatrick heat-bath dynamics tends to 1 as the system size n tends to infinity, both for the continuous-time dynamics and for the discrete-time dynamics. The model has spin configurations x in {±1}^n and disorder J consisting of independent standard Gaussian couplings J_{ij} for pairs i<j; the Hamiltonian is H(x)=(1/√n)·Σ_{i<j} J_{ij} x_i x_j, the Gibbs measure gives x weight exp(βH(x)) divided by the partition function, and the heat-bath kernel picks a uniformly random site i and resamples its spin from the conditional law exp(β y_i h_i)/(2cosh(βh_i)), where h_i is the local field (1/√n)·Σ_j J_{ij} x_j. The discrete kernel is the k-step power of this one-step kernel, and the continuous kernel at time t is the Poisson mixture Σ_k e^{-nt}(nt)^k/k! of the discrete kernels. The distance from a start x is the total variation distance between the kernel at that time and the Gibbs measure. With time scale exp(n^κ), κ=1/10000, the continuous bad mass is the Gaussian-disorder expectation of the total Gibbs mass of starting configurations x whose total variation distance at time exp(n^κ) exceeds 1/4, and the discrete bad mass is the same with k=⌊exp(n^κ)⌋ steps.