Fiche détaillée

Articles dans des revues avec comité de lecture

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

Liste des fichiers attachés à ce document :

PDF cycles.pdf(466.4 KB)

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

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.

hal-00598589, version 1

http://hal.archives-ouvertes.fr/hal-00598589

oai:hal.archives-ouvertes.fr:hal-00598589

Contributeur : Laurent Miclo <>

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

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