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 ) = 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 → ∞ . 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.