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

Sihan Wang

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arXiv · 2608.22159

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

Sihan Wang

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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.

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Sihan Wang. 2026-08-28. Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at $β< 1/2$. https://arxiv.org/abs/2608.22159

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