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

Author

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  • Daneshgar, Amir
  • Javadi, Ramin
  • Miclo, Laurent

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.

Suggested Citation

  • Daneshgar, Amir & Javadi, Ramin & Miclo, Laurent, 2012. "On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1748-1776.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1748-1776
    DOI: 10.1016/j.spa.2012.02.009
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