Abstract
We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time \({\frac{1}{2(1-\beta)}n\log n}\) with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n 3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n 3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β c = 1. In particular, we find a scaling window of order \({1/\sqrt{n}}\) around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ 2 n → ∞ with n, the mixing-time has order (n/δ) log(δ 2 n), and exhibits cutoff with constant \({\frac{1}{2}}\) and window size n/δ. In the critical window, β = 1± δ, where δ 2 n is O(1), there is no cutoff, and the mixing-time has order n 3/2. At low temperature, β = 1 + δ for δ > 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order \({\frac{n}{\delta}{\rm exp}\left((\frac{3}{4}+o(1))\delta^2n\right)}\).
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Communicated by H. Spohn
Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ding, J., Lubetzky, E. & Peres, Y. The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model. Commun. Math. Phys. 289, 725–764 (2009). https://doi.org/10.1007/s00220-009-0781-9
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DOI: https://doi.org/10.1007/s00220-009-0781-9