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Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability
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  • Published: 18 December 2008

Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability

  • David A. Levin1,
  • Malwina J. Luczak2 &
  • Yuval Peres3,4,5 

Probability Theory and Related Fields volume 146, Article number: 223 (2010) Cite this article

Abstract

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 − β)]−1 n log n. For β = 1, we prove that the mixing time is of order n 3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    David A. Levin

  2. Department of Mathematics, London School of Economics, Houghton Street, London, WC2A 2AE, UK

    Malwina J. Luczak

  3. Microsoft Research, Redmond, WA, USA

    Yuval Peres

  4. University of Washington, Seattle, WA, USA

    Yuval Peres

  5. University of California, Berkeley, CA, USA

    Yuval Peres

Authors
  1. David A. Levin
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  2. Malwina J. Luczak
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  3. Yuval Peres
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Corresponding author

Correspondence to David A. Levin.

Additional information

Research of Y. Peres was supported in part by NSF grant DMS-0605166.

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Levin, D.A., Luczak, M.J. & Peres, Y. Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146, 223 (2010). https://doi.org/10.1007/s00440-008-0189-z

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  • Received: 10 December 2007

  • Revised: 01 October 2008

  • Published: 18 December 2008

  • DOI: https://doi.org/10.1007/s00440-008-0189-z

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Keywords

  • Markov chains
  • Ising model
  • Curie–Weiss model
  • Mixing time
  • Cut-off
  • Coupling
  • Glauber dynamics
  • Metastability
  • Heat-bath dynamics
  • Mean-field model

Mathematics Subject Classification (2000)

  • 60J10
  • 60K35
  • 82C20
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