Combinatorial Solution of the Two-Matrix Model

TL;DR

This paper presents an exact solution to the two-matrix model using Schwinger-Dyson equations, enabling calculation of correlation functions involving angular degrees of freedom, with implications for lattice string theory and QCD.

Contribution

It introduces a novel elementary method to solve the two-matrix model, overcoming previous limitations in calculating angular correlations.

Findings

Exact solutions for angular correlation functions

Potential solvability of complex matrix models in physics

Insights into boundary conditions in 2D quantum gravity

Abstract

We write down and solve a closed set of Schwinger-Dyson equations for the two-matrix model in the large $N$ limit. Our elementary method yields exact solutions for correlation functions involving angular degrees of freedom whose calculation was impossible with previously known techniques. The result sustains the hope that more complicated matrix models important for lattice string theory and QCD may also be solvable despite the problem of the angular integrations. As an application of our method we briefly discuss the calculation of wavefunctions with general matter boundary conditions for the Ising model coupled to $2D$ quantum gravity. Some novel insights into the relationship between lattice and continuum boundary conditions are obtained.