A homological approach to the Gaussian Unitary Ensemble

TL;DR

This paper employs noncommutative geometry and homological methods to analyze the Gaussian Unitary Ensemble, deriving recurrence relations and generalizing Wigner's semicircle law for large matrix sizes.

Contribution

It introduces a homological framework using BV formalism to study GUE, leading to new recurrence relations and a generalized semicircle law.

Findings

Derived recurrence relations for GUE correlation functions

Generalized Wigner's semicircle law for large N

Computed large N statistical correlations for multi-trace functions

Abstract

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.