AI 中文总结
该研究针对β<1/2的Sherrington-Kirkpatrick模型，利用Bakry-Émery判据等建立最优阶庞加莱不等式，结合定位方案框架得到单站点Glauber动力学的最优混合时间界，核心思想来自GPT-5.6 Sol Ultra。
AI 中文摘要
我们证明，对于每一个固定的逆温度β<1/2，在无序的高概率下，n自旋Sherrington-Kirkpatrick模型的单站点Glauber动力学从任意初始构型出发，以总变差距离ε混合，需要O_β(n log(n/ε))步。该界在所有外场上一致成立，且仅依赖β的常数因子下是最优的。主要的关键是一般伊辛模型最优阶庞加莱不等式的确定性判据，通过整合Bakry-Émery判据及新的两自旋估计建立。随后应用Chen和Eldan的定位方案框架，将庞加莱不等式提升为修正对数索伯列夫不等式，得到最优混合时间界。庞加莱不等式证明的核心思想由GPT-5.6 Sol Ultra生成。
英文摘要
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.
CommentsRevised version; added related references and improved exposition
