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

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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Correspondence to David A. Levin.

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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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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