This work establishes novel, optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models using the powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree of $G$ on a local scale. Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case, (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, Stefankonic and Yin: PTRF 2017], (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on results in [Hayes: FOCS 2006], (d) They also allow us to refine the celebrated connection between the hardness of approximate counting and the phase transitions from statistical physics. We obtain our mixing bounds by utilising the $k$-non-backtracking matrix $H_{G,k}$. This is a very interesting, alas technically intricate, object to work with. We upper bound the spectral radius of the pairwise influence matrix $I^{\Lambda,\tau}_{G}$ by means of the 2-norm of $H_{G,k}$. To our knowledge, obtaining mixing bound using $H_{G,k}$ has not been considered before in the literature.

翻译：本研究利用[Anari, Liu, Oveis-Gharan: FOCS 2020]提出的谱独立性方法，为硬核模型与伊辛模型上的Glauber动力学建立了新颖且最优的混合时间界。这些界限通过底层图$G$的局部连接常数表示，该常数反映了图$G$在局部尺度上的有效度数。我们的结果对有界度图具有若干重要推论：(a) 将最大度数界作为特例包含其中；(b) 改进了[Sinclair, Srivastava, Stefankonic and Yin: PTRF 2017]中FPTAS算法的运行时间；(c) 能够获得基于邻接矩阵谱半径的混合时间界，并改进了[Hayes: FOCS 2006]中的结果；(d) 可进一步精确刻画近似计数计算复杂度与统计物理相变之间的经典关联。我们通过运用$k$阶非回溯矩阵$H_{G,k}$获得混合时间界，该矩阵虽技术处理复杂但具有重要理论价值。我们利用$H_{G,k}$的2-范数上界控制了成对影响矩阵$I^{\Lambda,\tau}_{G}$的谱半径。据我们所知，利用$H_{G,k}$获得混合时间界的研究在现有文献中尚未见报道。