Higher Order Random Walks, Local Spectral Expansion, and Applications

Abstract

The study of spectral expansion of graphs and expander graphs has been an extremely fruitful line of research in Mathematics and Computer Science, with applications ranging from random walks and fast sampling to optimization. In this dissertation, we study high dimensional local spectral expansion, which is a generalization of the theory of spectral expansion of graphs, to simplicial complexes.

We study two random walks on simplicial complexes, which we call the down-up walk, which captures a wide array of natural random walks which can be used to sample random combinatorial objects via the so-called heat-bath dynamics, and the swap walk, which can be thought as a random walk on a sparse version of the Kneser graph.

First, we give a sharp bound for the spectral gap of the down-up walks in terms of the local spectral expansion. Using this bound, we argue that the natural Markov chains for (i) sampling a random independent of fixed size s of a graph G = (V,E) is rapidly mixing, so long as s ≤ |V|/(∆+η) – where ∆ is the maximum degree of any vertex in G, and η is the magnitude of the least eigenvalue of the adjacency matrix of G; and (ii) sampling a common independent set from two partition matroids of fixed size s is rapidly mixing, so long as s ≤ r/3 – where r is the maximum size of any common independent set contained in both partition matroids.

Next, we study the spectrum of the swap walks, and show that using local spectral expansion we can relate the spectrum of the swap walk on any simplicial complex to the spectrum of the Kneser graph. We will mention applications of this result in (i) approximating constraint satisfaction problems (CSPs) on instances where the constraint hypergraph is a high dimensional local spectral expander; and in (ii) the construction of new families of list decodable codes based on (sparse) Ramanujan complexes of Lubotzky, Samuels, and Vishne.