On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes

Amir Daneshgar, Ramin Javadi and Laurent Miclo

Stochastic Processes and their Applications, 2012, vol. 122, issue 4, 1748-1776

Abstract: Let L be a reversible Markovian generator on a finite set V. Relations between the spectral decomposition of L and subpartitions of the state space V into a given number of components which are optimal with respect to min–max or max–min Dirichlet connectivity criteria are investigated. Links are made with higher-order Cheeger inequalities and with a generic characterization of subpartitions given by the nodal domains of an eigenfunction. These considerations are applied to generators whose positive rates are supported by the edges of a discrete cycle ZN, to obtain a full description of their spectra and of the shapes of their eigenfunctions, as well as an interpretation of the spectrum through a double-covering construction. Also, we prove that for these generators, higher Cheeger inequalities hold, with a universal constant factor 48.

Keywords: Reversible Markovian generator; Spectral decomposition; Cheeger’s inequality; Principal Dirichlet eigenvalues; Dirichlet connectivity spectra; Nodal domains of eigenfunctions; Optimal partitions of state space; Markov processes on discrete cycles (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spa.2012.02.009

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