Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at β< 1/2
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 ε in Oβ(n log(n / ε)) 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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BibTeX
@misc{wang2026optimal,
title = {{Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at $β< 1/2$}},
author = {Sihan Wang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.22159},
eprint = {2608.22159},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22159},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Mixing of Glauber Dynamics for the Sherrington-Kirkpatrick Model at β< 1/2
AU - Sihan Wang
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.22159
UR - https://arxiv.org/abs/2608.22159
AB - 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 ε in O_β(n log(n / ε)) 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.
ER -