Random matrices with external source and the asymptotic behaviour of multiple orthogonal polynomials

A. I. Aptekarev   ; V. G. Lysov   ; D. N. Tulyakov

Sbornik. Mathematics, Tome 202 (2011) no. 2, pp. 155-206

Résumé

Ensembles of random Hermitian matrices with a distribution measure defined by an anharmonic potential perturbed by an external source are considered. The limiting characteristics of the eigenvalue distribution of the matrices in these ensembles are related to the asymptotic behaviour of a certain system of multiple orthogonal polynomials. Strong asymptotic formulae are derived for this system. As a consequence, for matrices in this ensemble the limit mean eigenvalue density is found, and a variational principle is proposed to characterize this density. Bibliography: 35 titles.

Détail

Keywords: strong asymptotics, matrix Riemann-Hilbert problem, extremal problems in the theory of logarithmic potentials.
Mots-clés : random matrices, multiple orthogonal polynomials

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