具有任意边界条件的一般自旋系统的截断
Cutoff for general spin systems with arbitrary boundary conditions
摘要：
截断现象描述了马尔可夫链收敛到平衡点的急剧转变。在最近的工作中，作者建立了$d$ -维环面$(Z/nZ)^d$上随机伊辛模型的截断点及其位置，对于任意的$d\geq 1$ .该证明以本质上的方式使用了环面的对称结构和单调性。在这里，我们加强了框架，并将其扩展到一般的几何形状，边界条件和外部领域，推导出一个截止标准，涉及球的增长率和对数Sobolev常数的Glauber动力学。特别是，我们证明了有界度图的任何序列的次指数增长性在任意外部领域的逆对数Sobolev常数是有界的随机伊辛有截止。对于具有均匀边界的晶格，如全加，我们确定了明确的截断位置在半平面相交的无限体积动力学的谱隙。对于高温下的非单调自旋系统，包括气体硬核模型、Potts模型、反铁磁Potts模型和着色模型，得到了类似的建立截止的结果。
英文摘要：
The cutoff phenomenon describes a sharp transition in the convergence of a Markov chain to equilibrium. In recent work, the authors established cutoff and its location for the stochastic Ising model on the $d$-dimensional torus $(Z/nZ)^d$ for any $d\geq 1$. The proof used the symmetric structure of the torus and monotonicity in an essential way. Here we enhance the framework and extend it to general geometries, boundary conditions and external fields to derive a cutoff criterion that involves the growth rate of balls and the log-Sobolev constant of the Glauber dynamics. In particular, we show there is cutoff for stochastic Ising on any sequence of bounded-degree graphs with sub-exponential growth under arbitrary external fields provided the inverse log-Sobolev constant is bounded. For lattices with homogenous boundary, such as all-plus, we identify the cutoff location explicitly in terms of spectral gaps of infinite-volume dynamics on half-plane intersections. Analogous results establishing cutoff are obtained for non-monotone spin-systems at high temperatures, including the gas hard-core model, the Potts model, the anti-ferromagnetic Potts model and the coloring model.
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