A vector equilibrium problem for the two-matrix model in the quartic/quadratic case

TL;DR

This paper analyzes the asymptotic zero distribution of biorthogonal polynomials in a two-matrix model with specific potentials, revealing a vector equilibrium problem with novel critical behavior for negative parameter values.

Contribution

It introduces a new vector equilibrium problem characterizing the zero distribution and uncovers a novel critical behavior when the parameter t is negative.

Findings

Zero distribution characterized by a vector equilibrium problem.

Active external field on the third measure for t<0.

New critical behavior at negative t values.

Abstract

We consider the two sequences of biorthogonal polynomials (p_{k,n})_k and (q_{k,n})_k related to the Hermitian two-matrix model with potentials V(x) = x^2/2 and W(y) = y^4/4 + ty^2. From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p_{n,n} as n tends to infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t=0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behavior for a certain negative value of t. We also prove a general result about the interlacing of zeros of biorthogonal polynomials.