Conservation Laws of Random Matrix Theory

TL;DR

This paper reviews conservation laws in random matrix theory, deriving explicit formulas for matrix moments and genus expansion coefficients, which connect to combinatorics and quantum gravity, resolving longstanding conjectures.

Contribution

It introduces new conservation laws for matrix moments of Hermitean random matrices and provides explicit formulas for genus expansion coefficients, advancing understanding in mathematical physics.

Findings

Derived closed-form expressions for genus expansion coefficients.

Connected matrix moments to combinatorial enumeration of g-maps.

Resolved a 30+ year old conjecture in quantum gravity literature.

Abstract

This paper presents an overview of the derivation and significance of recently derived conservation laws for the matrix moments of Hermitean random matrices with dominant exponential weights that may be either even or odd. This is based on a detailed asymptotic analysis of the partition function for these unitary ensembles and their scaling limits. As a particular application we derive closed form expressions for the coefficients of the genus expansion for the associated free energy in a particular class of dominant even weights. These coefficients are generating functions for enumerating g-maps, related to graphical combinatorics on Riemann surfaces. This generalizes and resolves a 30+ year old conjecture in the physics literature related to quantum gravity.