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Uspekhi Mat. Nauk, 2011, Volume 66, Issue 6(402), Pages 123–190 (Mi umn9454)  

This article is cited in 44 scientific papers (total in 45 papers)

Hermite–Padé approximations and multiple orthogonal polynomial ensembles

A. I. Aptekareva, A. Kuijlaarsb

a M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow
b Katholieke Universiteit Leuven, Belgium

Abstract: This paper is concerned with Hermite–Padé rational approximants of analytic functions and their connection with multiple orthogonal polynomial ensembles of random matrices. Results on the analytic theory of such approximants are discussed, namely, convergence and the distribution of the poles of the rational approximants, and a survey is given of results on the distribution of the eigenvalues of the corresponding random matrices and on various regimes of such distributions. An important notion used to describe and to prove these kinds of results is the equilibrium of vector potentials with interaction matrices. This notion was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981.
Bibliography: 91 titles.

Keywords: Hermite–Padé approximants, multiple orthogonal polynomials, weak and strong asymptotics, extremal equilibrium problems for a system of measures, matrix Riemann–Hilbert problem, Christoffel–Darboux formula, matrix model with an external source, non-intersecting paths, two-matrix model.


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English version:
Russian Mathematical Surveys, 2011, 66:6, 1133–1199

Bibliographic databases:

UDC: 517.53
MSC: Primary 41A21, 42C05, 60B20; Secondary 31A15, 60G17, 60G55
Received: 15.09.2011

Citation: A. I. Aptekarev, A. Kuijlaars, “Hermite–Padé approximations and multiple orthogonal polynomial ensembles”, Uspekhi Mat. Nauk, 66:6(402) (2011), 123–190; Russian Math. Surveys, 66:6 (2011), 1133–1199

Citation in format AMSBIB
\by A.~I.~Aptekarev, A.~Kuijlaars
\paper Hermite--Pad\'e approximations and multiple orthogonal polynomial ensembles
\jour Uspekhi Mat. Nauk
\yr 2011
\vol 66
\issue 6(402)
\pages 123--190
\jour Russian Math. Surveys
\yr 2011
\vol 66
\issue 6
\pages 1133--1199

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