Asymptotic optimality of isoperimetric constants with respect to $L^{2}(\pi)$-spectral gaps

TL;DR

This paper studies the conditions under which Markov chains on general state spaces have spectral gaps in the $L^{2}(\pi)$ sense, linking these gaps to isoperimetric constants and their asymptotic behavior, with implications for mixing times.

Contribution

It provides necessary and sufficient conditions for the existence of $L^{2}(\pi)$-spectral gaps using isoperimetric constants and explores their asymptotic properties, offering new insights into spectral analysis of Markov chains.

Findings

Spectral gaps characterized by isoperimetric constants.

Asymptotic behavior of spectral gaps established.

Connections between spectral gaps and convergence to uniform flow.

Abstract

In this paper we investigate the existence of $L^{2}(\pi)$-spectral gaps for $\pi$-irreducible, positive recurrent Markov chains on general state space. We obtain necessary and sufficient conditions for the existence of $L^{2}(\pi)$-spectral gaps in terms of a sequence of isoperimetric constants and establish their asymptotic behavior. It turns out that in some cases the spectral gap can be understood in terms of convergence of an induced probability flow to the uniform flow. The obtained theorems can be interpreted as mixing results and yield sharp estimates for the spectral gap of some Markov chains.