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.

One of the key components in PCP constructions are agreement tests. In agreement test the tester is given access to subsets of fixed size of some set, each equipped with an assignment. The tester is then tasked with testing whether these local assignments agree with some global assignment over the entire set. One natural generalization of this concept is the case where, instead of a single assignment to each local view, the tester is given access to $l$ different assignments for every subset. The tester is then tasked with testing whether there exist $l$ global functions that agree with all of the assignments of all of the local views. In this work we present sufficient condition for a set system to exhibit this generalized definition of list agreement expansion. This is, to our knowledge, the first work to consider this natural generalization of agreement testing. Despite initially appearing very similar to agreement expansion, list agreement expansion seem to require a different set of techniques. This is due to the fact that the natural extension of agreement testing does not suffice when testing for list agreement, as list agreement crucially relies on a global structure. It follows that if a local assignments satisfy list agreement they must not only agree locally but also exhibit some additional structure. In order to test for the existence of this additional structure we use a connection between covering spaces of a high dimensional complex and its coboundaries. We use this connection as a form of ``decoupling''. Moreover, we show that any set system that exhibits list agreement expansion also supports direct sum testing. This is the first scheme for direct sum testing that works regardless of the parity of the sizes of the local sets. Prior to our work the schemes for direct sum testing were based on the parity of the sizes of the local tests.

In this work, using methods from high dimensional expansion, we show that the property of k-direct-sum is testable for odd values of k . Previous work of [Kaufman and Lubotzky, 2014] could inherently deal only with the case that k is even, using a reduction to linearity testing. Interestingly, our work is the first to combine the topological notion of high dimensional expansion (called co-systolic expansion) with the combinatorial/spectral notion of high dimensional expansion (called colorful expansion) to obtain the result. The classical k-direct-sum problem applies to the complete complex; Namely it considers a function defined over all k-subsets of some n sized universe. Our result here applies to any collection of k-subsets of an n-universe, assuming this collection of subsets forms a high dimensional expander.