GUE Correlators and Large Genus Asymptotics

TL;DR

The paper investigates the large-genus asymptotics of GUE correlators, which encode combinatorial counts of ordinary and ribbon graphs via a matrix-resolvent formula. It provides explicit representations for normalized graph counts as genus grows, proving a universal leading limit of 1 and demonstrating that these counts depend rationally on the genus, with precise $1/g$ expansions. By connecting GUE observables to graph enumeration through tau-function and resolvent techniques, the results illuminate the structure of large-genus behavior and rationality properties in random matrix models and their geometric interpretations. The methods extend prior work on large-genus asymptotics and offer concrete, computable asymptotic expansions for both ordinary and 1-face ribbon graphs.

Abstract

In this paper, we use a formula obtained in [8] to study certain asymptotic behaviors of GUE (Gaussian unitary ensemble) correlators. More precisely, we obtain large genus asymptotics of enumerations of ordinary graphs and ribbon graphs with 1 face.

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 4
• Corollary 2
• Lemma 1
• Lemma 2
• proof : Proof of Theorem \ref{['thmOG']}
• proof : Proof of Theorem \ref{['thm2OG']}