Combinatorics of dispersionless integrable systems and universality in random matrix theory

TL;DR

This paper demonstrates the universality of leading order free energies in large random matrix ensembles through dispersionless integrable systems, and derives explicit formulas connecting ribbon graphs and Faber polynomials.

Contribution

It explicitly computes leading free energies for Gaussian ensembles and shows their universality across different matrix ensembles using dispersionless integrable hierarchies.

Findings

Leading orders of free energies agree across ensembles.

Explicit formula for two-point function $F_{nm}$.

Connection between ribbon graphs and Faber polynomials.

Abstract

It is well-known that the partition function of the unitary ensembles of random matrices is given by a tau-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are tau-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function $F_{nm}$ which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and $\frac{1}{nm} F_{nm}$ are the Grunsky coefficients of the Faber polynomials.