22066 articles – 15901 references  [version française]
 Short view Article in peer-reviewed journal
 On nodal domains of finite reversible Markov processes and spectral decomposition of cyclesDaneshgar A. et alStochastic Processes and Applications 122, 4 (2012) 1748--1776 - http://hal.archives-ouvertes.fr/hal-00598589
Attached file list to this document:
 PDF
 cycles.pdf(466.4 KB)
Amir Daneshgar1, Ramin Javadi1, Laurent Miclo ()2
 1: Department of Mathematical Sciences http://mathsci.sharif.ir/ Sharif University of Technology Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran Iran, Islamic Republic Of 2: IMT - Institut de Mathématiques de Toulouse 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 Bâtiment 1R3 118 route de Narbonne 31062 TOULOUSE CEDEX 4 France
 Mathematics/Probability Mathematics/Functional Analysis
On nodal domains of finite reversible Markov processes and spectral decomposition of cycles
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.
English
2011-06-07

Stochastic Processes and Applications
international
2012
122
4
1748--1776

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.
60J27, 60J35, 60E15, 15A18, 49K35, 31C20, 05C38
24 pages

 hal-00598589, version 1 http://hal.archives-ouvertes.fr/hal-00598589 oai:hal.archives-ouvertes.fr:hal-00598589 From: Laurent Miclo <> Submitted on: Tuesday, 7 June 2011 08:53:44 Updated on: Thursday, 6 December 2012 21:35:08