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：高维扩张图最重要的特性之一是高维随机游走能够快速收敛。这一性质在计算机科学理论的多个领域中被证明极其有用，涵盖了一致性测试、采样、编码理论等。本文在前人分析高维随机游走收敛性的工作（引用~\cite{DBLP:conf/innovations/KaufmanM17,DBLP:conf/focs/DinurK17, DBLP:conf/approx/KaufmanO18,DBLP:journals/corr/abs-2001-02827}）基础上，通过提出文献~\cite{DBLP:journals/corr/abs-2001-02827}结果的**结构化**版本，给出了该领域的最新成果。以往研究从最坏特征值的角度考察扩张性，而本文则利用函数的结构，将函数的扩张性与随机游走算子的整个谱联系起来；我们将此类定理称为细粒度高阶随机游走定理。在具有充分结构的情况下，本文提出的细粒度结果可能远优于最坏情形；而在最坏情形下，我们的结果与文献~\cite{DBLP:journals/corr/abs-2001-02827}等价。为了证明细粒度高阶随机游走定理，我们引入了一种方法：在扩张性足够好的前提下，将复形顶点上随机游走的扩张性提升为对高阶随机游走的细粒度理解。此外，我们的**单一**提升定理可以同时推导出细粒度高阶随机游走定理和著名的滴落定理。在此之前，高阶随机游走定理与滴落定理需通过不同的证明方法获得。