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

2007/12/05 by Levin, David A., Luczak, Malwina J., Peres, Yuval · 4 citations
#60J10 #60K35 #82C20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0712.0790

Abstract

We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 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-beta)]^-1 n log n. For beta = 1, we prove that the mixing time is of order n^3/2. For beta > 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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• Fast mixing in Ising models with a negative spectral outlier via Gaussian approximation
• Limit Profile for the high-temperature Curie-Weiss model
• Polynomial mixing of the critical Ising model on sparse Erdos-Renyi graphs
• Cutoff for the Swendsen-Wang dynamics on the complete graph

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