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Random matrices with external source and multiple orthogonal polynomials

P. M. Bleher·A. B. J. Kuijlaars
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Abstract

We show that the average characteristic polynomial P_n(z) = E [\det(zI-M)] of the random Hermitian matrix ensemble Z_n^{-1} \exp(-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a_j of A, there is a weight and P_n has n_j orthogonality conditions with respect to this weight, if n_j is the multiplicity of a_j. 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.

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P. M. Bleher, A. B. J. Kuijlaars. 2003-07-28. Random matrices with external source and multiple orthogonal polynomials. https://arxiv.org/abs/math-ph/0307055

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