Related papers: High-Dimensional Expanders from Expanders
Random walks on regular bounded degree expander graphs have numerous applications. A key property of these walks is that they converge rapidly to the uniform distribution on the vertices. The recent study of expansion of high dimensional…
Random walks on bounded degree expander graphs have numerous applications, both in theoretical and practical computational problems. A key property of these walks is that they converge rapidly to their stationary distribution. In this work…
We study high order random walks in high dimensional expanders; namely, in complexes which are local spectral expanders. Recent works have studied the spectrum of high order walks and deduced fast mixing. However, the spectral gap of high…
We study a variant of the down-up and up-down walks over an $n$-partite simplicial complex, which we call expanderized higher order random walks -- where the sequence of updated coordinates correspond to the sequence of vertices visited by…
We give a new rapid mixing result for a natural random walk on the independent sets of a graph $G$. We show that when $G$ has bounded treewidth, this random walk -- known as the Glauber dynamics for the hardcore model -- mixes rapidly for…
We present a new construction of high dimensional expanders based on covering spaces of simplicial complexes. High dimensional expanders (HDXs) are hypergraph analogues of expander graphs. They have many uses in theoretical computer…
The cutoff phenomenon describes a sharp transition in the convergence of an ergodic finite Markov chain to equilibrium. Of particular interest is understanding this convergence for the simple random walk on a bounded-degree expander graph.…
High-dimensional expanders generalize the notion of expander graphs to higher-dimensional simplicial complexes. In contrast to expander graphs, only a handful of high-dimensional expander constructions have been proposed, and no elementary…
Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add…
Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass [KM16], yet their broader combinatorial and spectral properties…
The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second…
Many complex systems exhibit interactions that depend not only on pairwise connections, but also group structures and memory effects. To capture such effects, we develop a unified tensor framework for modeling higher-order Markov chains…
We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA,…
We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to…
Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…
Random walks on a graph reflect many of its topological and spectral properties, such as connectedness, bipartiteness and spectral gap magnitude. In the first part of this paper we define a stochastic process on simplicial complexes of…
There is a long history of establishing central limit theorems for Markov chains. Quantitative bounds for chains with a spectral gap were proved by Mann and refined later. Recently, rates of convergence for the total variation distance were…
We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using…
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…
It is well known that the spectral gap of the down-up walk over an $n$-partite simplicial complex (also known as Glauber dynamics) cannot be better than $O(1/n)$ due to natural obstructions such as coboundaries. We study an alternative…