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On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes

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  • Daneshgar, Amir
  • Javadi, Ramin
  • Miclo, Laurent
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    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 generic 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 ZN, 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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 4 ()
    Pages: 1748-1776

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1748-1776

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    Related research

    Keywords: 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;

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