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The mixing time evolution of Glauber dynamics for the mean-field Ising model. (English) Zbl 1173.82018
Glauber dynamics of the Ising model on the complete graph is considered. The authors obtained a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point \(\beta_c=1\). A scaling window of order \(1/\sqrt{n}\) around the critical temperature is found. It is shown that in the high temperature regime, \(\beta=1-\delta\) for some \(0<\delta<1\) so that \(\delta^2n\to\infty\) with \(n\), the mixing-time has order \((n/\delta)\log(\delta^2n)\), and exhibits cutoff with constant \(1/2\) and window size \(n/\delta\). In the critical window, \(\beta=1\pm \delta\), where \(\delta^2n\) is \(o(1)\), there is no cutoff, and the mixing-time has order \(n^{3/2}\). At low temperature, \(\beta=1+\delta\) for \(\delta>0\) with \(\delta^2n\to\infty\) and \(\delta=o(1),\) there is no cutoff, and the mixing time has order \({n\over \delta}\exp(({3\over 4}+o(1))\delta^2n)\).

MSC:
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
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