Fisher zeros and correlation decay in the Ising model
Fisher zeros and correlation decay in the Ising model
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DOI：
10.1063/1.5082552
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发表时间：
2018-07
期刊：
影响因子：
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通讯作者：
Jingcheng Liu;A. Sinclair;P. Srivastava
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文献类型：
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作者：
Jingcheng Liu;A. Sinclair;P. Srivastava
伊辛模型起源于统计物理学，作为研究磁体相变的一种手段，近世纪来一直是深入研究的对象。从组合上看，它可以被看作是图中割集上的自然分布，并且在计算机科学中也得到了广泛的研究，特别是在近似计数和采样的背景下。在本文中，我们研究了伊辛模型的配分函数的复零点，被视为“相互作用参数”中的多项式;这些被称为Fisher零点，因为它们是由Fisher在1965年引入的。虽然自李和杨的经典工作以来，配分函数作为“场”参数的多项式的零点已被广泛研究，但对Fisher零点知之甚少。我们的主要结果表明，零场伊辛模型在整个参数区域的复杂邻域中没有Fisher零点，模型表现出相关衰减。除了阐明费歇尔零点本身之外，这个结果还建立了伊辛模型相变的两个不同概念之间的正式联系：不存在复零点（自由能的解析性，或配分函数的对数）和相关性随距离的衰减。我们还讨论了我们的结果的后果，有效的确定性近似的配分函数。我们的证明在很大程度上依赖于算法技术，特别是韦茨的自我避免步行树，因此属于越来越多的工作，使用算法的方法来解决统计物理学中的经典问题。
The Ising model originated in statistical physics as a means of studying phase transitions in magnets, and has been the object of intensive study for almost a century. Combinatorially, it can be viewed as a natural distribution over cuts in a graph, and it has also been widely studied in computer science, especially in the context of approximate counting and sampling. In this paper, we study the complex zeros of the partition function of the Ising model, viewed as a polynomial in the "interaction parameter"; these are known as Fisher zeros in light of their introduction by Fisher in 1965. While the zeros of the partition function as a polynomial in the "field" parameter have been extensively studied since the classical work of Lee and Yang, comparatively little is known about Fisher zeros. Our main result shows that the zero-field Ising model has no Fisher zeros in a complex neighborhood of the entire region of parameters where the model exhibits correlation decay. In addition to shedding light on Fisher zeros themselves, this result also establishes a formal connection between two distinct notions of phase transition for the Ising model: the absence of complex zeros (analyticity of the free energy, or the logarithm of the partition function) and decay of correlations with distance. We also discuss the consequences of our result for efficient deterministic approximation of the partition function. Our proof relies heavily on algorithmic techniques, notably Weitz's self-avoiding walk tree, and as such belongs to a growing body of work that uses algorithmic methods to resolve classical questions in statistical physics.