Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
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深淬火后的相序：树上的随机 Ising 和硬核气体模型
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
F. Martinelli
中科院分区：
文献类型：
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作者：
P. Caputo;F. Martinelli
考虑具有马尔可夫半群的相共存区域中的低温随机伊辛模型。一个基本且在很大程度上仍然悬而未决的问题是，当从高度无序的状态δη(例如乘积伯努利测量或高温吉布斯测量)采样初始组态η时，对ν铂的长时间行为的理解。利用正则二元树上离散自旋模型的蒙特卡罗马尔可夫链混合时间分析的最新进展，我们研究了Ising模型和硬核气体(独立集)模型的上述问题。如果ν是有偏乘积伯努利定律，那么，在关于偏差和热力学参数的各种假设下，我们证明了ν-几乎必然弱收敛到极值吉布斯测度(纯相)，并且证明了极限至少和时间t的拉伸指数一样快地接近极限。在随机化算法的背景下，如果考虑大的有限树上的Glauber动力学，我们的结果证明在比真实混合时间小得多的时间尺度上快速局部松弛到平衡，只要链的起点不被认为是最坏的，而是从合适的分布中采样。
Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup Pt. A fundamental and still largely open problem is the understanding of the long time behavior of δηPt when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree , we study the above problem for the Ising and hard core gas (independent sets) models on . If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δηPt to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.