Mixed correlation function and spectral curve for the 2-matrix model

TL;DR

This paper develops new methods to compute mixed correlation functions and spectral curves in the 2-matrix model, confirming a conjecture that links the spectral curve to loop equations.

Contribution

It introduces a novel approach involving orthogonal polynomials near degree n and derives new formulas for spectral curves and differential systems in the 2-matrix model.

Findings

Mixed correlation functions expressed via orthogonal polynomials.

Spectral curve matches the one in loop equations, confirming Bertola's conjecture.

New representations for differential systems of biorthogonal polynomials.

Abstract

We compute the mixed correlation function in a way which involves only the orthogonal polynomials with degrees close to $n$, (in some sense like the Christoffel Darboux theorem for non-mixed correlation functions). We also derive new representations for the differential systems satisfied by the biorthogonal polynomials, and we find new formulae for the spectral curve. In particular we prove the conjecture of M. Bertola, claiming that the spectral curve is the same curve which appears in the loop equations.