发表机构
Nanjing University（南京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Sherrington-Kirkpatrick模型的Glauber动力学采样，提出基于弱Poincaré不等式的多项式时间模拟退火算法，实现逆多项式精度，并绕过Talagrand开放问题获得配分函数估计。
AI 中文摘要
我们研究使用Glauber动力学从Sherrington-Kirkpatrick (SK)模型的Gibbs分布中进行采样。对于每个固定的逆温度$0\leq\beta<1$和每个固定的$M,D>0$，我们给出一个多项式时间的模拟退火算法，其输出分布与Gibbs分布的total-variation距离在$n^{-M}$以内，且对相互作用矩阵而言概率至少为$1-n^{-D}$。该算法从均匀乘积自旋开始，并沿递增的逆温度调度使用Glauber动力学。同样的方法还产生相对误差为$n^{-M}$的配分函数估计。关键成分是一个定量的弱Poincaré不等式。基于随机局部化方法处理弱Poincaré不等式[Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026]，我们使用高斯等周不等式直接比较精确的随机局部化路径。我们的比较通过控制后验均值的局部Lipschitz连续性的累积成本来容忍其失败。局部Lipschitz性来自于在近似消息传递定位的局部强凸区域中，将后验均值近似为Thouless-Anderson-Palmer (TAP)自由能的驻点，这一范式由算法随机局部化引入[El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]。近似误差大致由TAP梯度方程的有效性刻画，但在不同的坐标中。为了证明误差很少累积过多，我们需要强集中控制。直接论证需要解决Talagrand [2010, Research Problem 1.7.9]的一个开放问题的变体。我们通过一个中间条件化步骤绕过这个开放问题，并通过建立TAP梯度在删除坐标下的稳定性将控制转移回来。
英文摘要
We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leqβ<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.
