Peierls等值线的损失网络表示
Loss network representation of Peierls contours
摘要：
我们提出了一个概率的方法研究系统的排除，在传统上通过集群扩展方法研究的制度。本文着重讨论它在低温下研究伊辛模型中发现的Peierls等值线的气体中的应用，但大多数结果是一般性的。我们实现的平衡测度的不变测度的损失网络过程的存在是由一个占主导地位的分支过程的亚临界条件。在这种情况下，该方法产生，除了存在性和唯一性的措施，性能，如指数空间收敛和混合，和一个中心极限定理。损失网络指数快速收敛到平衡测度，没有亚稳态陷阱。这种收敛速度在低温下更快，在那里它导致证明的渐近泊松分布的轮廓。我们的混合性质的措施的结果是可比的“重复变量扩展”，用于治疗系统的混乱和耦合的地图格子。它比通常的团簇膨胀形式主义在更大的有效区域中工作，并且它不依赖于压力的解析性。事实上，它不会导致后者的任何形式的扩展，并且平衡测度的性质是在不诉诸组合或复分析技术的情况下获得的。
英文摘要：
We present a probabilistic approach for the study of systems with exclusions, in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss-network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime, the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with ``duplicated-variables expansion'', used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.
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