Duality of spectral curves arising in two-matrix models

TL;DR

This paper studies the spectral properties of two-matrix models, deriving dual systems of ODEs for biorthonormal polynomials and proving their spectral curve equivalence in the large size limit.

Contribution

It introduces a method to derive dual finite systems of ODEs for biorthonormal polynomials and proves their spectral curve equivalence as the matrix size approaches infinity.

Findings

Dual systems of ODEs are derived for biorthonormal polynomials.

Spectral curves of dual systems are shown to coincide in the large N limit.

An inverse theorem reconstructs measures from recursion relations and string equations.

Abstract

The two matrix model is considered, with measure given by the exponential of a sum of polynomials in two different variables. It is shown how to derive a sequence of pairs of ``dual'' finite size systems of ODEs for the corresponding biorthonormal polynomials. An inverse theorem is proved showing how to reconstruct such measures from pairs of semi-infinite finite band matrices defining the recursion relations and satisfying the string equation. A proof is given in the $N\to \infty$ limit that the dual systems obtained share the same spectral curve.