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：本文为谱独立命题提供了自成体系的讲义。对于$n$顶点图，谱独立条件是对$n\times n$影响矩阵最大特征值的约束，该矩阵元素刻画顶点对间的相互影响，且与协方差矩阵密切相关。我们将展示近期成果：谱独立性保证Glauber动力学的混合时间为多项式（多项式次数取决于特定参数）。证明过程运用了局部到全局定理，本讲义将详细阐述这些定理。最后，我们将介绍更前沿的成果：谱独立性可导出松弛时间（逆谱隙）的最优界，并在附加条件下得到Glauber动力学混合时间的最优界$O(n\log{n})$。同时呈现Anari、Liu、Oveis Gharan与Vinzant（2019）关于拟阵随机基生成的成果。相关基交换游走的分析，借助Oppenheim（2018）的滴流定理与谱独立性的局部到全局定理共同分析局部游走。本讲义重点从泛函分析视角研究相关马尔可夫链的谱隙，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。