OAI.CriticalSK.critical_mixing_bounds

The theorem states that for every ε>0 the following two limits equal 1, for the Sherrington–Kirkpatrick-type spin system defined in the context file. For n spins x∈{±1}ⁿ, the disorder W assigns a real coupling to each pair i<j, drawn independently from a centered Gaussian with variance 1/n (the law disorderLaw n). The energy is H(x)=Σ_{i<j} W_{ij}xᵢxⱼ, with no extra inverse-temperature parameter, and the Gibbs distribution is exp(H(x)) divided by the sum of exp(H) over all configurations. The site kernel for coordinate i resamples spin i from the Gibbs distribution conditioned on the other spins. The discrete kernel averages these n site kernels with weight 1/n each, the generator is the sum over i of (site kernel i minus identity), and the continuous-time kernel at time t is the matrix exponential exp(tL) of the generator. Distance is total variation (half the ℓ¹ distance) to the Gibbs distribution. The continuous mixing time is the infimum of t≥0 such that the distance from every starting configuration is at most 1/4, and the discrete mixing time is the least natural number k with this property for the kernel's kth power. The first limit says that the probability, under the disorder law, that n^{2/3−ε} ≤ continuous mixing time ≤ e^{εn} tends to 1 as n→∞. The second says that the probability that n^{5/3−ε} ≤ discrete mixing time ≤ e^{εn} also tends to 1. The result is stated in the source with its proof omitted (sorry).