Biorthogonal polynomials for 2-matrix models with semiclassical potentials

TL;DR

This paper develops a comprehensive framework for biorthogonal polynomials in two-matrix models with rational potentials and hard-edges, deriving recurrence relations, differential equations, and Riemann-Hilbert problems to analyze their properties.

Contribution

It introduces new recurrence relations, Christoffel-Darboux identities, and a Riemann-Hilbert formulation for biorthogonal polynomials with semiclassical potentials in two-matrix models.

Findings

Derived explicit recurrence relations depending on hard-edges.

Established Christoffel-Darboux identities for these polynomials.

Formulated a Riemann-Hilbert problem capturing polynomial properties.

Abstract

We consider the biorthogonal polynomials associated to the two-matrix model where the eigenvalue distribution has potentials V_1,V_2 with arbitrary rational derivative and whose supports are constrained on an arbitrary union of intervals (hard-edges). We show that these polynomials satisfy certain recurrence relations with a number of terms d_i depending on the number of hard-edges and on the degree of the rational functions V_i'. Using these relations we derive Christoffel-Darboux identities satisfied by the biorthogonal polynomials: this enables us to give explicit formulae for the differential equation satisfied by d_i+1 consecutive polynomials, We also define certain integral transforms of the polynomials and use them to formulate a Riemann-Hilbert problem for (d_i+1) x (d_i+1) matrices constructed out of the polynomials and these transforms. Moreover we prove that the Christoffel-Darboux pairing can be interpreted as a pairing between two dual Riemann-Hilbert problems.