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

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

Amir Daneshgar 1, Ramin Javadi 1, Laurent Miclo 2

(2012)

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
	Sharif University of Technology
2 :	Institut de Mathématiques de Toulouse (IMT)
	Université Paul Sabatier [UPS] - Toulouse III – Université Toulouse le Mirail - Toulouse II – Université des Sciences Sociales - Toulouse I – Institut National des Sciences Appliquées (INSA) - Toulouse – CNRS : UMR5219

Domaine

:

Mathématiques/Probabilités Mathématiques/Analyse fonctionnelle

Mots Clés : 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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Contributeur : Laurent Miclo <>

Soumis le : Mardi 7 Juin 2011, 08:53:44

Dernière modification le : Jeudi 6 Décembre 2012, 21:35:08