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

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).

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