平均场Blume-Capel模型在聚集路径耦合下的混合时间
Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling
摘要：
本文研究了统计力学系统Glauber动力学的混合时间与热力学平衡结构之间的关系。对于这一点，我们考虑平均场的布卢姆-卡佩尔模型，最简单的统计力学模型，表现出以下复杂的相变结构：在一个二维参数空间内存在一条曲线，在该模型经历二阶，连续相变，曲线，在该模型经历一阶，不连续相变，和三临界点，这两条曲线分开。我们确定的缓慢和快速混合的区域之间的接口。为了完全确定区域的快速混合，我们采用了一种新的扩展的路径耦合方法，成功地证明了快速混合，即使在相邻状态之间没有收缩。
英文摘要：
In this paper we investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.
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