Random matrices with external source and multiple orthogonal polynomials

We show that the average characteristic polynomial Pn(z) = E [det(zI−M)] of the random Hermitian matrix ensemble Zn−1exp(−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...

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Bibliographic Details
Published in:International Mathematics Research Notices 2004, Vol.2004 (3), p.109-129
Main Authors: Bleher, P. M., Kuijlaars, A. B. J.
Format: Article
Language:English
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Summary:We show that the average characteristic polynomial Pn(z) = E [det(zI−M)] of the random Hermitian matrix ensemble Zn−1exp(−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 (1997). 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.
ISSN:1073-7928
1687-1197
1687-0247
DOI:10.1155/S1073792804132194