二维Swendsen-Wang动力学的快速混合

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2012-12-20 DOI:10.4064/dm502-0-1

Mario Ullrich

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引用次数: 37

摘要

我们在任意图上证明了Swendsen-Wang (SW)动力学、Potts模型的热浴(HB)动力学和随机簇模型的单键(SB)动力学的比较结果。特别地，我们证明了在最大度有界的图上，HB的快速混合意味着SW的快速混合，并且SW的快速混合与SB的快速混合是等价的。此外，SW和SB在平面图上的谱隙由上到下由相应的对偶图上的谱隙限定，并适当地改变温度。因此，我们得到二维方形晶格上Potts模型的Swendsen-Wang动力学在所有非临界温度下的快速混合，以及二维Ising模型在所有温度下的快速混合。此外，在足够高或足够低的温度下，我们得到了一般图的新结果。

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Rapid mixing of Swendsen–Wang dynamics in two dimensions

We prove comparison results for the Swendsen-Wang (SW) dynamics, the heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics for the random-cluster model on arbitrary graphs. In particular, we prove that rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum degree and that rapid mixing of SW and rapid mixing of SB are equivalent. Additionally, the spectral gap of SW and SB on planar graphs is bounded from above and from below by the spectral gap of these dynamics on the corresponding dual graph with suitably changed temperature. As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the Potts model on the two-dimensional square lattice at all non-critical temperatures as well as rapid mixing for the two-dimensional Ising model at all temperatures. Furthermore, we obtain new results for general graphs at high or low enough temperatures.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.