Title: Random matrices with external source and multiple orthogonal polynomials
Authors: Bleher, PM ×
Kuijlaars, Arno #
Issue Date: 2004
Series Title: International mathematics research notices vol:2004 issue:3 pages:109-129
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.
ISSN: 1073-7928
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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