Lectures on the Asymptotic Expansion of a Hermitian Matrix Integral
Skip to main content
Open Access Publications from the University of California

Department of Mathematics

Faculty bannerUC Davis

Lectures on the Asymptotic Expansion of a Hermitian Matrix Integral

  • Author(s): Mulase, Motohico
  • et al.

Published Web Location

No data is associated with this publication.

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.

Item not freely available? Link broken?
Report a problem accessing this item