论文标题
图形Zeta功能和Kazakov-Migdal模型中的Wilson Loop
Graph Zeta Functions and Wilson Loops in Kazakov-Migdal Model
论文作者
论文摘要
在本文中，我们考虑了在任意图上定义的扩展的Kazakov-Migdal模型。该模型的分区函数表示为图表上所有Wilson循环的求和，事实证明是由Bartholdi Zeta函数表示，该函数由图形边缘上的单位矩阵加权。在有限$ n $处的周期图上的分区功能由广义加泰罗尼亚数字的生成函数表示。可以在大的$ n $上精确评估任意图上的分区函数，该函数以一种变形的ihara zeta函数表示为无限的产物。 Wilson Loops的非零区域没有助长自由能的1/N $扩展的领先部分，而是对下一个领先的领先部分。标量场特征值的半圆分布仍然是该模型在任意常规图上大的$ n $上的精确解决方案，但仅反映了零区域Wilson循环。
In this paper, we consider an extended Kazakov-Migdal model defined on an arbitrary graph. The partition function of the model, which is expressed as the summation of all Wilson loops on the graph, turns out to be represented by the Bartholdi zeta function weighted by unitary matrices on the edges of the graph. The partition function on the cycle graph at finite $N$ is expressed by the generating function of the generalized Catalan numbers. The partition function on an arbitrary graph can be exactly evaluated at large $N$ which is expressed as an infinite product of a kind of deformed Ihara zeta function. The non-zero area Wilson loops do not contribute to the leading part of the $1/N$-expansion of the free energy but to the next leading. The semi-circle distribution of the eigenvalues of the scalar fields is still an exact solution of the model at large $N$ on an arbitrary regular graph, but it reflects only zero-area Wilson loops.
