Potential Hessian Ascent III: Sampling the Sherrington--Kirkpatrick Model at Beta < 1/2

2026-05-05

AI Summary

This work addresses the problem of efficiently sampling from the Gibbs distribution of the antiferromagnetic Sherrington–Kirkpatrick model in the high-temperature regime (inverse temperature β < 1/2). To this end, the authors propose a stochastic localization sampler that integrates Hessian ascent on a potential function with Glauber dynamics. This method achieves, for the first time, efficient exact sampling in total variation distance (TVD) across the entire replica-symmetric phase (β < 1/2), overcoming prior limitations that either restricted applicability to β ≈ 0.295 or controlled only Wasserstein error. By synthesizing tools from Gaussian integration by parts, overlap concentration, the cavity method, free probability theory, the Jarzynski equality, and restricted log-Sobolev inequalities, the algorithm attains o(1) TVD error in polynomial time while maintaining O(1) Wasserstein and KL divergence errors.

Abstract

We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington--Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature $β< 1/2$. Prior work obtained TVD error guarantees only up to $β\approx 0.295$, while results covering the entire replica-symmetric regime $β< 1$ gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an $O(1)$ Wasserstein error guarantee for finite-time localization, improving on the previous $o(n)$. A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling the diagonal sub-algebra of the Hessian improves this to $O(1)$ in KL divergence. We then use Jarzynski's equality with rejection sampling, along with a restricted log-Sobolev inequality on the time-$T$ localized distribution, to refine the error to $o(1)$ in TVD up to a constant time $T$ and to complete the sampling with Glauber dynamics.

Problem

• Sherrington–Kirkpatrick model
• Gibbs sampling
• total-variation distance
• inverse temperature
• replica-symmetric regime

Innovation

• Potential Hessian Ascent
• Stochastic Localization
• Sherrington–Kirkpatrick Model
• Total Variation Distance
• Gibbs Sampling