Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics for the Sherrington-Kirkpatrick model up to $β\leq \frac{1}{2}+\varepsilon$

2026-09-11

AI Summary

本文通过引入一种新的局部到全局技术来解决Glauber动力学在Sherrington-Kirkpatrick模型中的混合时间问题，利用精心选择的秩-1扰动改进了影响矩阵的上界估计。

Abstract

We introduce a new family of trickledown theorems, a.k.a., local to global technique to bound the spectral gap of the Glauber dynamics for multi-state spin systems. In this technique instead of upper-bounding the influence matrix of a link of co-dimension 2 by $λI$ (where $λ$ is the second eigenvalue of the link), we upper-bound the influence matrix after a carefully chosen rank-1 shift. The rank-1 shift allows for a significantly smaller upper-bound but it comes at the cost of bounding the average loss due to rank-1 perturbations. As an application we use this method to show that the natural Glauber dynamics mixes in polynomial time to generate samples from the Sherrington-Kirkpatrick model for $β\leq \tfrac{1}{2}+\varepsilon$, for an absolute constant $\varepsilon>0$. At the heart of the proof we manage to bound the loss due to rank-1 perturbations by averaging over all links of co-dimension 2.

Problem

• Glauber dynamics
• Sherrington-Kirkpatrick model
• mixing time
• spectral gap

Innovation

• rank-1-perturbed trickledown theorems
• spectral gap
• Glauber dynamics
• Sherrington-Kirkpatrick model