格点上Ising模型的截断
Cutoff for the Ising model on the lattice
摘要：
介绍于1963年，格劳伯动力学是一个最实践和广泛研究的方法，采样的伊辛模型的格子。众所周知，在高温下，该链在尺寸为$n$的体系上混合到$L^1$中所花费的时间为$O(\log n)$。在这种情况下，是否存在截断，即$L^1$ -收敛到平衡的急剧转变，是一个基本的开放问题：如果是这样，正如Peres所解释的，这意味着对于某些固定的$c>0$，混合在$(c+o(1))\log n$突然发生，从而为MCMC采样器提供了严格的停止规则。然而，即使对于相当简单的马尔可夫链，获得精确的混合渐近性和证明截断也是非常具有挑战性的。对于一维伊辛模型，显示截止是一个长期的开放问题。我们解决了上述问题，建立截止和它的位置在高温制度的伊辛模型的晶格周期性边界条件。我们的结果适用于任何维度和任何温度，在那里有很强的空间混合：对于$\Z^2$，这进行所有的方式到临界温度。具体来说，对于固定的$d\geq 1$，具有周期性边界条件的$(\Z/n\Z)^d$上Ising模型的连续时间Glauber动力学在$(d/2λ_\infty)\log n$处截止，其中$λ_\infty$是无限体积晶格上动力学的谱间隙。据我们所知，这是第一次在马尔可夫链的截止显示，甚至了解其平稳分布是有限的。证明铰链上的一个新的技术转换$L^1$的$L^2$混合，使应用程序的log-Sobolev不等式。该方法具有普遍性，可推广到其它单调和反单调自旋系统。
英文摘要：
Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in $L^1$ on a system of size $n$ is $O(\log n)$. Whether in this regime there is cutoff, i.e. a sharp transition in the $L^1$-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at $(c+o(1))\log n$ for some fixed $c>0$, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For $\Z^2$ this carries all the way to the critical temperature. Specifically, for fixed $d\geq 1$, the continuous-time Glauber dynamics for the Ising model on $(\Z/n\Z)^d$ with periodic boundary conditions has cutoff at $(d/2λ_\infty)\log n$, where $λ_\infty$ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating $L^1$ to $L^2$ mixing which enables the application of log-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems.
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