Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at $β< 1/2$

2026-08-22

AI Summary

该研究证明了在特定条件下，单点Glauber动力学可以在O(n log(n/ε))步内解决Sherrington-Kirkpatrick模型的最优混合问题。

Abstract

We prove that for every fixed inverse temperature $β< 1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_β\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $β$. The main ingredient is a deterministic criterion for optimal-order Poincaré inequalities in general Ising models, established via the integrated Bakry-Émery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincaré inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincaré inequality were generated by GPT-5.6 Sol Ultra.

Problem

• Glauber Dynamics
• Sherrington-Kirkpatrick Model
• Mixing Time
• Total Variation Distance

Innovation

• Glauber Dynamics
• Sherrington-Kirkpatrick Model
• Poincaré Inequality
• Modified Log-Sobolev Inequality