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

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.