| Author(s) |
Amir Daneshgar1, Ramin Javadi1, Laurent Miclo ( )2 |
| Laboratory |
| 1: |
Department of Mathematical Sciences |
 |
| http://mathsci.sharif.ir/ |
|
| Sharif University of Technology |
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| 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 |
|
| Subject |
|
| Title |
On nodal domains of finite reversible Markov processes and spectral decomposition of cycles |
| 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 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. |
| Fulltext language |
English |
| Production date |
2011-06-07 |
|
| Journal |
Stochastic Processes and Applications |
| Audience |
international |
| Publication date |
2012 |
| Volume |
122 |
| Issue |
4 |
| Page, identifiant, ... |
1748--1776 |
|
| 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. |
| Classification |
60J27, 60J35, 60E15, 15A18, 49K35, 31C20, 05C38 |
| Comment |
24 pages |
|
