Isoperimetric Constants and Spectral Properties for Time Reversible Markov Chains

#### A. Wubker

2011, v.17, №4, 483-498

ABSTRACT

In this paper we study the spectral properties of Markov operators on $L^{2}$-spaces. Lawler and Sokal [Trans. Amer. Math. Soc., 1988, 309, pp. 557-580] used isoperimetric constants for discrete and continuous time Markov chains to obtain a spectral gap at one. For discrete time Markov chains this does not exclude periodic behavior. We define a new constant measuring the distance from periodicity and give necessary and sufficient conditions for the existence of a global spectral gap in terms of this constant.

Keywords: spectral gap,isoperimetric constant,$L^2$-spaces,reversibility,Markov chain