Simplicial complexes: spectrum, homology and random walks

TL;DR

This paper extends the concepts of spectral graph theory and random walks to higher-dimensional simplicial complexes, exploring their topological properties, spectral gaps, and behaviors in infinite complexes.

Contribution

It introduces a stochastic process on simplicial complexes that captures higher homology and spectral properties, generalizing classical graph results to higher dimensions.

Findings

Defined a higher-dimensional random walk reflecting homology.

Generalized spectral gap results to infinite complexes.

Identified conditions under which high-dimensional spectral gap theorems hold.

Abstract

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 arbitrary dimension, which reflects in an analogue way the existence of higher dimensional homology, and the magnitude of the high-dimensional spectral gap originating in the works of Eckmann and Garland. The second part of the paper is devoted to infinite complexes. We present a generalization of Kesten's result on the spectrum of regular trees, and of the connection between return probabilities and spectral radius. We study the analogue of the Alon-Boppana theorem on spectral gaps, and exhibit a counterexample for its high-dimensional counterpart. We show, however, that under some assumptions the theorem does hold - for example, if the codimension-one skeletons of the complexes in question form a family of expanders. Our study suggests natural generalizations of many concepts from graph theory, such as amenability, recurrence/transience, and bipartiteness. We present some observations regarding these ideas, and several open questions.