Censored Glauber Dynamics for the Mean Field Ising Model
Censored Glauber Dynamics for the Mean Field Ising Model
复制标题
平均场伊辛模型的截尾格劳伯动力学
DOI：
10.1007/s10955-009-9859-1
复制
发表时间：
2008
影响因子：
1.6
通讯作者：
Y. Peres
中科院分区：
文献类型：
--
作者：
Jian Ding;E. Lubetzky;Y. Peres
本文研究了n阶完全图上Ising模型（Curie-Weiss模型）的Glauber动力学。众所周知，在高温（β<1）下，混合时间为Θ（nlog n），而在低温（β>1）下，混合时间为exp（Θ（n））。最近，Levin，Luczak和Peres考虑了这种动力学的删失版本，该版本仅限于非负磁化。他们证明，对于固定的β>1，该模型的混合时间是Θ（nlog n），类似于原始动力学的高温状态。此外，它们显示了固定β<1的原始动力学的截止值。删失动力学是否也表现出截断的问题仍然没有解决。在一篇配套论文中，我们将Levin等人的结果推广到Curie-Weiss模型的混合时间的完整表征。也就是说，我们找到了一个有序的缩放窗口，
$1/\sqrt{n}$
在临界温度βc=1附近，超过该临界温度，在高温下存在截止。然而，确定临界窗口之外的删失动力学行为似乎更具挑战性，本文对上述问题给予了肯定的回答，并建立了临界窗口之外的删失动力学的截断点及其窗口，从而完成了它与高温下原始动力学的类比。也就是说，若β=1+δ，且δ>0，δ2n→∞，则混合时间为（n/δ）log（δ2n）阶.截止常数是（1/2+[2（β 2/δ−1）]−1），其中β是g（x）=tanh（βx）−x的唯一正根，截止窗口的阶数为n/δ。
AbstractWe study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled.In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order
$1/\sqrt{n}$
around the critical temperature βc=1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging.In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ2n→∞, then the mixing-time has order (n/δ)log (δ2n). The cutoff constant is (1/2+[2(ζ2β/δ−1)]−1), where ζ is the unique positive root of g(x)=tanh (βx)−x, and the cutoff window has order n/δ.