Random matrices with external source and multiple orthogonal polynomials

2003/07/28 by P. M. Bleher, A. B. J. Kuijlaars · 1 citation
Physics and Astronomy · Mathematics · #math-ph #hep-th #math.CA #math.MP

paper · pdf

published as Int.Math.Res.Not.2004:109-129,2004 · 17 pages

arxiv created 2003/07/28 · arxiv updated 2011/03/28

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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