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

We consider the two sequences of biorthogonal polynomials and 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 p_n,n as n → ∞. 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 behaviour for a certain negative value of t.

We also prove a general result about the interlacing of zeros of biorthogonal polynomials.