The Z-invariant Ising model via dimers
The Z-invariant Ising model via dimers
复制标题
通过二聚体的 Z 不变 Ising 模型
DOI：
10.1007/s00440-018-0861-x
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发表时间：
2016
影响因子：
2
通讯作者：
K. Raschel
中科院分区：
文献类型：
--
作者：
Cédric Boutillier;B. Tilière;K. Raschel
Z 不变 Ising 模型（Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315–346, 1978）在等径图上定义，并且具有取决于椭圆参数 k 的耦合常数。当 k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document} 模型为至关重要，并且随着 k 的变化，整个温度范围都被覆盖。在本文中，我们研究了 Fisher 图上相应的二聚体模型，从而将我们的论文（Boutillier 和 de Tilière in Probab Theory Relat Fields 147:379–413, 2010；Commun Math Phys 301(2):473–516, 2011）扩展到完全 Z 不变的情况。我们的主要结果之一是 Kasteleyn 算子的逆的显式局部公式。它最显着的特点是它是 Bouutillier 和 de Tilière (2011) 的椭圆推广：它涉及到局部函数和 Bouutilier 等人引入的大规模离散指数函数。 （发明数学 208(1):109–189, 2017）。这特别表明 Z 不变性，而不是关键性，是获得局部表达式的核心。然后，我们计算渐进并推导出自然吉布斯测度的显式局部表达式。我们证明了伊辛模型自由能的局部公式。我们还证明，在常数范围内，这种自由能与 Bouutilier 等人的 Z 不变跨越森林的自由能相等。 (2017)，并推断出两个模型在 k 上具有相同的二阶相变。接下来，我们证明该模型的自对偶关系，将 Baxter 的结果扩展到所有等径图。在最后一部分中，我们证明二分图上二聚体模型的显式局部表达式，对应于该 Z 不变伊辛模型的 XOR 版本。
The Z-invariant Ising model (Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315–346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter k. When k=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=0$$\end{document} the model is critical, and as k varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers (Boutillier and de Tilière in Probab Theory Relat Fields 147:379–413, 2010; Commun Math Phys 301(2):473–516, 2011) to the fullZ-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of Boutillier and de Tilière (2011): it involves a local function and the massive discrete exponential function introduced in Boutillier et al. (Invent Math 208(1):109–189, 2017). This shows in particular that Z-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the Z-invariant spanning forests of Boutillier et al. (2017), and deduce that the two models have the same second order phase transition in k. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this Z-invariant Ising model.