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Evolving sets, mixing and heat kernel bounds
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  • Published: 06 June 2005

Evolving sets, mixing and heat kernel bounds

  • B. Morris1 &
  • Yuval Peres2 

Probability Theory and Related Fields volume 133, pages 245–266 (2005)Cite this article

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Abstract

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pn(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California, Evans Hall, Berkeley, CA, 94720, USA

    B. Morris

  2. Departments of Statistics and Mathematics, University of California, Berkeley, USA

    Yuval Peres

Authors
  1. B. Morris
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  2. Yuval Peres
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Corresponding author

Correspondence to B. Morris.

Additional information

Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.

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Cite this article

Morris, B., Peres, Y. Evolving sets, mixing and heat kernel bounds. Probab. Theory Relat. Fields 133, 245–266 (2005). https://doi.org/10.1007/s00440-005-0434-7

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  • Received: 25 May 2003

  • Revised: 17 January 2005

  • Published: 06 June 2005

  • Issue Date: October 2005

  • DOI: https://doi.org/10.1007/s00440-005-0434-7

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Keywords

  • Markov Chain
  • State Space
  • Nash
  • Stationary Distribution
  • Mathematical Biology
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