Random matrices with external source and multiple orthogonal polynomials
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.
Cite
@article{arxiv.math-ph/0307055,
title = {Random matrices with external source and multiple orthogonal polynomials},
author = {P. M. Bleher and A. B. J. Kuijlaars},
journal= {arXiv preprint arXiv:math-ph/0307055},
year = {2011}
}
Comments
17 pages