Lectures on the Asymptotic Expansion of a Hermitian Matrix Integral

TL;DR

This paper presents three rigorous methods for computing the asymptotic expansion of Hermitian matrix integrals, revealing new solutions to KP equations and connecting combinatorics, orthogonal polynomials, and integrable systems.

Contribution

It introduces a combinatorial, analytical, and integrable systems approach to asymptotic expansions, including novel transcendental solutions to KP equations.

Findings

Combinatorial method links to Riemann surface tilings.

Orthogonal polynomial analysis yields asymptotics involving the Riemann zeta function.

New transcendental solutions to KP equations are discovered.

Abstract

In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann $\zeta$-function. The third method is derived from a formula for the $\tau$-function solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are \emph{transcendental}, in the sense that they cannot be obtained by the celebrated Krichever construction and its generalizations based on algebraic geometry of vector bundles on Riemann surfaces. In each case a mathematically rigorous way of dealing with asymptotic series in an infinite number of variables is established.