One of the most important properties of high dimensional expanders is that high dimensional random walks converge rapidly. This property has proven to be extremely useful in variety of fields in the theory of computer science from agreement testing to sampling, coding theory and more. In this paper we present a state of the art result in a line of works analyzing the convergence of high dimensional random walks~\cite{DBLP:conf/innovations/KaufmanM17,DBLP:conf/focs/DinurK17, DBLP:conf/approx/KaufmanO18,DBLP:journals/corr/abs-2001-02827}, by presenting a \emph{structured} version of the result of~\cite{DBLP:journals/corr/abs-2001-02827}. While previous works examined the expansion in the viewpoint of the worst possible eigenvalue, in this work we relate the expansion of a function to the entire spectrum of the random walk operator using the structure of the function; We call such a theorem a Fine Grained High Order Random Walk Theorem. In sufficiently structured cases the fine grained result that we present here can be much better than the worst case while in the worst case our result is equivalent to~\cite{DBLP:journals/corr/abs-2001-02827}. In order to prove the Fine Grained High Order Random Walk Theorem we introduce a way to bootstrap the expansion of random walks on the vertices of a complex into a fine grained understanding of higher order random walks, provided that the expansion is good enough. In addition, our \emph{single} bootstrapping theorem can simultaneously yield our Fine Grained High Order Random Walk Theorem as well as the well known Trickling down Theorem. Prior to this work, High order Random walks theorems and Tricking down Theorem have been obtained from different proof methods.

翻译：高维扩展器最重要的性质之一是高维随机游走能够快速收敛。该性质已在计算机科学理论的众多领域中被证明极其有用，从一致性测试到采样、编码理论等。本文通过提出一个结构化版本的[引用文献~\cite{DBLP:journals/corr/abs-2001-02827}]结果，呈现了分析高维随机游走收敛性研究系列中最先进的结果。先前的研究从最差可能特征值的角度考察扩展性，而本文则利用函数的结构，将函数的扩展性与随机游走算子的整个谱相关联；我们将此定理称为细粒度高阶随机游走定理。在充分结构化的情形下，本文提出的细粒度结果可能远优于最差情形，而在最差情形下，我们的结果与~[引用文献~\cite{DBLP:journals/corr/abs-2001-02827}]等价。为证明细粒度高阶随机游走定理，我们引入了一种方法：将复形顶点上随机游走的扩展性引导为对高阶随机游走的细粒度理解，前提是扩展性足够好。此外，我们的单一引导定理能够同时导出细粒度高阶随机游走定理和著名的渗流定理。在此工作之前，高阶随机游走定理与渗流定理需通过不同的证明方法获得。