秩一微扰滴落定理：$\beta\leq \frac{1}{2}+\varepsilon$ 时 Sherrington-Kirkpatrick 模型 Glauber 动力学的混合时间
Rank-1-perturbed trickledown theorems: Mixing time of Glauber dynamics for the Sherrington-Kirkpatrick model up to $β\leq \frac{1}{2}+\varepsilon$
查看机构详情
- University of Washington（华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。
浏览论文内容
中文总结 AI 辅助
本文提出秩一微扰滴落定理，通过秩一移位改进谱间隙上界，证明 SK 模型在 $\beta\leq 1/2+\varepsilon$ 时 Glauber 动力学多项式时间混合。
中文摘要 AI 辅助
我们引入了一族新的滴落定理，即一种用于界定多态自旋系统 Glauber 动力学谱间隙的局部到整体技术。在该技术中，我们不再用 $\lambda I$（其中 $\lambda$ 是链接的第二特征值）来上界余维为 2 的链接的影响矩阵，而是经过精心选择的秩一移位后对该影响矩阵进行上界界定。秩一移位允许显著更小的上界，但代价是需要界定由秩一扰动引起的平均损失。作为应用，我们使用该方法证明了：对于绝对常数 $\varepsilon>0$，当 $\beta\leq \tfrac{1}{2}+\varepsilon$ 时，自然的 Glauber 动力学在多项式时间内混合，从而能够从 Sherrington-Kirkpatrick 模型中生成样本。证明的核心在于，我们通过对所有余维为 2 的链接取平均，成功地界定了由秩一扰动引起的损失。
英文摘要
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.
