These are self-contained lecture notes for spectral independence. For an $n$-vertex graph, the spectral independence condition is a bound on the maximum eigenvalue of the $n\times n$ influence matrix whose entries capture the influence between pairs of vertices, it is closely related to the covariance matrix. We will present recent results showing that spectral independence implies the mixing time of the Glauber dynamics is polynomial (where the degree of the polynomial depends on certain parameters). The proof utilizes local-to-global theorems which we will detail in these notes. Finally, we will present more recent results showing that spectral independence implies an optimal bound on the relaxation time (inverse spectral gap) and with some additional conditions implies an optimal mixing time bound of $O(n\log{n})$ for the Glauber dynamics. We also present the results of Anari, Liu, Oveis Gharan, and Vinzant (2019) for generating a random basis of a matroid. The analysis of the associated bases-exchange walk utilizes the local-to-global theorems used for spectral independence with the Trickle-Down Theorem of Oppenheim (2018) to analyze the local walks. Our focus in these notes is on the analysis of the spectral gap of the associated Markov chains from a functional analysis perspective, and we present proofs of the associated local-to-global theorems from this same Markov chain perspective.

翻译：这些是自成体系的谱独立性讲座笔记。对于$n$顶点图，谱独立性条件是对$n\times n$影响矩阵最大特征值的约束，该矩阵元素刻画了顶点对之间的影响关系，与协方差矩阵密切相关。我们将介绍最新研究成果，表明谱独立性蕴含格劳伯动力学的混合时间为多项式（多项式次数取决于特定参数）。本笔记将详细阐述证明中使用的局部到全局定理。最后，我们将展示更近期的结果：谱独立性可推导出弛豫时间（谱间隙的倒数）的最优界，并在附加条件下为格劳伯动力学提供$O(n\log{n})$的最优混合时间界。同时介绍Anari、Liu、Oveis Gharan与Vinzant（2019年）关于生成拟阵随机基的研究成果。相关基交换游走的分析运用了谱独立性中的局部到全局定理，并结合Oppenheim（2018年）的涓流定理分析局部游走。本笔记重点从泛函分析视角分析相关马尔可夫链的谱间隙，并从相同马尔可夫链视角给出相应局部到全局定理的证明。