On nodal domains and higher-order Cheeger inequalities of finite reversible Markov processes [electronic resource]

Daneshgar, A. (Amir) ; Sharif Univercity of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/j.spa.2012.02.009
  3. Abstract:
  4. Let LL be a reversible Markovian generator on a finite set View the MathML sourceV. Relations between the spectral decomposition of LL and subpartitions of the state space View the MathML sourceV 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 ZNZN, 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
  5. Keywords:
  6. Reversible Markovian generator ; Markov processes on discrete cycles ; Optimal partitions of state space ; Nodal domains of eigenfunctions ; Dirichlet connectivity spectra ; Principal Dirichlet eigenvalues ; Cheeger’s inequality ; Spectral decomposition
  7. Source: Stochastic Processes and their Applications ; Volume 122, Issue 4, April 2012, Pages 1748–1776
  8. URL: http://www.sciencedirect.com/science/article/pii/S0304414912000312