We present novel results for fast mixing of Glauber dynamics using the newly introduced and powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. We mainly focus on the Hard-core model and the Ising model. We obtain bounds for fast mixing with the parameters expressed in terms of the spectral radius of the adjacency matrix, improving on the seminal work in [Hayes: FOCS 2006]. Furthermore, we go beyond the adjacency matrix and establish -- for the first time -- rapid mixing results for Glauber dynamics expressed in terms of the spectral radius of the Hashimoto non-backtracking matrix of the underlying graph $G$. Working with the non-backtracking spectrum is extremely challenging, but also more desirable. Its eigenvalues are less correlated with the high-degree vertices than those of the adjacency matrix and express more accurately invariants of the graph such as the growth rate. Our results require ``weak normality" from the Hashimoto matrix. This condition is mild and allows us to obtain very interesting bound. We study the pairwise influence matrix ${I}^{\Lambda,\tau}_{G}$ by exploiting the connection between the matrix and the trees of self-avoiding walks, however, we go beyond the standard treatment of the distributional recursions. The common framework that underlies our techniques we call the topological method. Our approach is novel and gives new insights into how to establish Spectral Independence for Gibbs distributions. More importantly, it allows us to derive new -- improved -- rapid mixing bounds for Glauber dynamics on distributions such as the Hard-core model and the Ising model for graphs that the spectral radius is smaller than the maximum degree.

翻译：我们利用 [Anari, Liu, Oveis-Gharan: FOCS 2020] 中最新引入的强大谱独立性方法，提出了关于 Glauber 动力学快速混合的新结果。主要关注硬核模型和伊辛模型。我们以邻接矩阵谱半径形式表达参数，获得了快速混合的界，改进了 [Hayes: FOCS 2006] 中的开创性工作。此外，我们超越了邻接矩阵，首次建立了以底层图 $G$ 的 Hashimoto 非回溯矩阵谱半径表达的 Glauber 动力学快速混合结果。处理非回溯谱极具挑战性，但更具优势。与邻接矩阵相比，其特征值与高度数顶点的相关性更弱，能更准确地表达图的增长速率等不变量。我们的结果要求 Hashimoto 矩阵满足“弱正态性”。这一条件温和且能导出非常有趣的界。我们通过挖掘成对影响矩阵 ${I}^{\Lambda,\tau}_{G}$ 与自回避行走树之间的联系展开研究，但超越了分布递推的标准处理方法。我们将支撑这些技术的通用框架称为拓扑方法。该方法具有创新性，为建立吉布斯分布的谱独立性提供了新见解。更重要的是，它使我们能够针对谱半径小于最大度的图，为硬核模型和伊辛模型等分布推导出新的、更优的 Glauber 动力学快速混合界。