A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
Abstract
We consider the two sequences of biorthogonal polynomials (pk,n)k and (qk,n)k related to the Hermitian two-matrix model with potentials V(x) = x2/2 and W(y) = y4/4 + ty2. 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 pn,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.