X iv : m at h - ph / 0 30 70 55 v 1 2 8 Ju l 2 00 3 Random matrices with external source and multiple orthogonal polynomials
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چکیده
We show that the average characteristic polynomial P n (z) = E[det(zI−M)] of the random Hermitian matrix ensemble Z −1 n 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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