Random matrix model with external source and a constrained vector equilibrium problem

Abstract: We consider the random matrix model with external source, in case where the potential V(x) is an even polynomial and the external source has two eigenvalues a, -a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures (mu_1,mu_2) that solve a constrained vector equilibrium problem. The proof is based on the steepest descent analysis of the associated Riemann-Hilbert problem for multiple orthogonal polynomials. We illustrate our results in detail for the case of a quartic double well potential V(x) = x^4/4 - tx^2/2. We are able to determine the precise location of the phase transitions in the ta-plane, where either the constraint becomes active, or the two intervals in the support come together (or both).