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

Amir Daneshgar (1) , Ramin Javadi (1) , Laurent Miclo (2)

1 Department of Mathematical Sciences
2 IMT - Institut de Mathématiques de Toulouse UMR5219

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.

Mots clés: exit times; 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.

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Amir Daneshgar, Ramin Javadi, Laurent Miclo. On nodal domains of finite reversible Markov processes and spectral decomposition of cycles. Stochastic Processes and their Applications, 2012, 122 (4), pp.1748--1776. ⟨hal-00598589⟩