Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory |
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Authors: | M Vanlessen |
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Institution: | (1) Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium and Fakultat fur Mathematik, Ruhr-Universitat Bochum, Universitatsstrasse 150, 44801 Bochum, Germany |
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Abstract: | We consider polynomials orthogonal on 0,∞) with
respect to Laguerre-type weights w(x) = xα e-Q(x),
where α > -1 and where Q denotes a polynomial with
positive leading coefficient. The main purpose of this paper
is to determine Plancherel-Rotach-type asymptotics in the
entire complex plane for the orthonormal polynomials with
respect to w, as well as asymptotics of the corresponding
recurrence coefficients and of the leading coefficients of
the orthonormal polynomials. As an application we will use
these asymptotics to prove universality results in random
matrix theory. We will prove our results by using the characterization of
orthogonal polynomials via a 2 × 2 matrix valued
Riemann--Hilbert problem, due to Fokas, Its, and Kitaev,
together with an application of the Deift-Zhou steepest
descent method to analyze the Riemann-Hilbert problem
asymptotically. |
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Keywords: | Orthogonal polynomials Asymptotic analysis Riemann-Hilbert problems Random matrices |
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