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Stochastic Processes and Applications 122, 4 (2012) 1748--1776

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On nodal domains of finite reversible Markov processes and spectral decomposition of cycles

Amir Daneshgar1, Ramin Javadi1, Laurent Miclo2

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 generical 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 $\mathbf{Z}_N$, 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.

1:	Department of Mathematical Sciences
2:	IMT - Institut de Mathématiques de Toulouse

Keyword(s) : 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 – exit times.

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From: Laurent Miclo <>

Submitted on: Tuesday, 7 June 2011 08:53:44

Updated on: Thursday, 6 December 2012 21:35:08