圆柱的匹配与熵
Matchings and entropies of cylinders
摘要：
图的完美匹配的计数等价于在统计物理中有应用的二聚体问题。一个图$G$称为$n$ -旋转对称的，如果$n$阶的循环群是$G$的自同构群的子群. Jockusch（完美匹配和完满平方，J. Combin. Theory Ser.A，67（1994），100-115）和Kuperberg（An exploration of the permanent-determinant method，Electron.J.Combin.，5（1998），#46）独立证明了：如果$G$是一个具有$2n$ -旋转对称的$N$阶平面二部图，则$G$的完美匹配数可以表示为$N/2n$阶$n$行列式的乘积.在本文中，我们给这个结果一个新的介绍。我们利用这个结果计算了具有$2n$符号对称性的体平面二分格的熵。我们得到了两种类型的圆柱的完美匹配和熵的数量的显式表达式。利用Kenyon、Okounkov和Sheffield关于环面熵的结果，（Dimers and amoeulum，Ann.Math.163（2006），1019- 1056）以及Salinas和Nagle（层状氢键$SnCl^2\cdot 2H_2O$晶体中的相变理论，物理评论B，9（1974），4920- 4931），我们证明了每个考虑的圆柱和它相应的环面具有相同的熵。最后，我们提出了一些问题。
英文摘要：
The enumeration of perfect matchings of graphs is equivalent to the dimer problem which has applications in statistical physics. A graph $G$ is said to be $n$-rotation symmetric if the cyclic group of order $n$ is a subgroup of the automorphism group of $G$. Jockusch (Perfect matchings and perfect squares, J. Combin. Theory Ser. A, 67(1994), 100-115) and Kuperberg (An exploration of the permanent-determinant method, Electron. J. Combin., 5(1998), #46) proved independently that if $G$ is a plane bipartite graph of order $N$ with $2n$-rotation symmetry, then the number of perfect matchings of $G$ can be expressed as the product of $n$ determinants of order $N/2n$. In this paper we give this result a new presentation. We use this result to compute the entropy of a bulk plane bipartite lattice with $2n$-notation symmetry. We obtain explicit expressions for the numbers of perfect matchings and entropies for two types of cylinders. Using the results on the entropy of the torus obtained by Kenyon, Okounkov, and Sheffield (Dimers and amoebae, Ann. Math. 163(2006), 1019--1056) and by Salinas and Nagle (Theory of the phase transition in the layered hydrogen-bonded $SnCl^2\cdot 2H_2O$ crystal, Phys. Rev. B, 9(1974), 4920--4931), we show that each of the cylinders considered and its corresponding torus have the same entropy. Finally, we pose some problems.
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