$L^{2}$-spectral gaps for time discrete reversible Markov chains

Achim Wuebker

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arXiv · 0908.0897

$L^{2}$-spectral gaps for time discrete reversible Markov chains

Achim Wuebker

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Abstract

In this paper we study the spectral properties of Markov-operator 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 1. For time discrete 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.

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Achim Wuebker. 2009-08-06. $L^{2}$-spectral gaps for time discrete reversible Markov chains. https://arxiv.org/abs/0908.0897

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