论文标题
高斯统一合奏的一种同源方法
A homological approach to the Gaussian Unitary Ensemble
论文作者
论文摘要
我们使用非交通性几何形状和Batalin-Vilkovisky（BV）形式主义的同源框架研究高斯单一合奏（GUE）。与等级$ n $相关函数的相关函数的系数用功能区图Feynman图描述了，然后导致相应表面的计数问题。此同源设置提供的规范关系决定了这些相关函数的复发关系。使用此复发关系和加泰罗尼亚数字的属性，我们确定相关函数相对于等级$ n $的领先顺序行为。作为应用程序，我们证明了Wigner的半圆定律的概括，并计算了由多条件函数定义的随机变量家族的所有大$ n $统计相关性。
We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the Batalin-Vilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are described in terms of ribbon graph Feynman diagrams that then lead to a counting problem for the corresponding surfaces. The canonical relations provided by this homological setup determine a recurrence relation for these correlation functions. Using this recurrence relation and properties of the Catalan numbers, we determine the leading order behavior of the correlation functions with respect to the rank $N$. As an application, we prove a generalization of Wigner's semicircle law and compute all the large $N$ statistical correlations for the family of random variables in the GUE defined by multi-trace functions.
