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Zbl 0956.42014
Bleher, Pavel; Its, Alexander
Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. (English)
[J] Ann. Math. (2) 150, No.1, 185-266 (1999). ISSN 0003-486X; ISSN 1939-0980/e

There is a revolution sweeping through asymptotics of orthogonal polynomials called the Riemann-Hilbert method. It has enabled researchers such as the present authors and Deift, Kriecherbauer, MacLaughlin and others to obtain very precise (and uniform) asymptotics for orthogonal polynomials for exponential weights, in situations where the classical Bernstein-Szegö methods give limited precision. And this paper is the record of one of the first breakthroughs in this exciting development.\par Let $$V(z):= gz^4/4+ tz^2/2,$$ where $g>0>t$, so that $V$ is a double-well potential. Let $0<\varepsilon< 1$, and for $n\ge 1$ consider a parameter $N$ satisfying $$\varepsilon>{n\over N}< {t^2\over 4g}- \varepsilon.$$ Let us consider the monic orthogonal polynomials $P_n$ with respect to the varying weight $w:= \exp(-NV)$, so that $$\int^\infty_{-\infty} P_nP_m\exp(- NV)= h_n\delta_{mn},$$ where $h_n> 0$. The authors establish very precise asymptotics for $P_n$ and the associated recurrence coefficients as $n\to\infty$. Then they apply these to establish universality of the local distribution of eigenvalues in the matrix model with quartic potential.\par A key point in the analysis is the Fokas-Its-Kitaev Riemann-Hilbert problem, in which the orthogonal polynomials appear explicitly. This is followed by use of an approximate solution to the Riemann-Hilbert problem, and a proof that the approximate solution gives the asymptotic formula. The paper contains an extensive review of related literature; in particular, the context of the results and their motivation is very clearly presented. This paper will be of great use to anyone interested in orthogonal polynomials and their applications.
[D.S.Lubinsky (Wits)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
33C05 Classical hypergeometric functions
15A52 Random matrices
33C45 Orthogonal polynomials and functions of hypergeometric type
41A60 Asymptotic problems in approximation

Keywords: random matrices; asymptotics; orthogonal polynomials; exponential weights; Riemann-Hilbert problem

Cited in: Zbl 1061.30035

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