Random matrices with external source and multiple orthogonal polynomials

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We show that the average characteristic polynomial Pₙ(z) = E [(zI-M)] of the random Hermitian matrix ensemble Zₙ⁻¹ (-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue aⱼ of A, there is a weight and Pₙ has nⱼ orthogonality conditions with respect to this weight, if nⱼ is the multiplicity of aⱼ. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.