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

Abstract

We prove that for every fixed inverse temperature $\beta<1 / 2$, 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 $O_{\beta}\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $\beta$. The main ingredient is a deterministic criterion for optimal-order Poincar\'e inequalities in general Ising models, established via the integrated Bakry-\'Emery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincar\'e inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincar\'e inequality were generated by GPT-5.6 Sol Ultra.