Mixed Correlation Functions of the Two-Matrix Model

TL;DR

This paper develops a method to compute mixed correlation functions of two non-commuting random matrices by reducing the problem to biorthogonal polynomial construction and determinant evaluation.

Contribution

It advances the calculation of mixed correlation functions in the two-matrix model by explicitly constructing biorthogonal polynomials and deriving their generating functions.

Findings

Derived explicit formulas for mixed correlation functions

Reduced the problem to biorthogonal polynomial recursion coefficients

Expressed generating functions as determinants involving these coefficients

Abstract

We compute the correlation functions mixing the powers of two non-commuting random matrices within the same trace. The angular part of the integration was partially known in the literature: we pursue the calculation and carry out the eigenvalue integration reducing the problem to the construction of the associated biorthogonal polynomials. The generating function of these correlations becomes then a determinant involving the recursion coefficients of the biorthogonal polynomials.