Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0944.42013
Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X.
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. (English)
[J] Commun. Pure Appl. Math. 52, No.11, 1335-1425 (1999). ISSN 0010-3640

In this important and interesting paper, the authors again illustrate the awesome power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Let $V: \bbfR\to\bbfR$ be real-valued and analytic on $\bbfR$, with $$\lim_{|x|\to\infty} V(x)/\log(1+ x^2)=\infty.$$ For $n\ge 1$, let $\{p_k(x; n)\}^\infty_{k= 0}$ denote the sequence of orthonormal polynomials for the weight $\exp(-nV)$, so that $$\int p_k(x; n)p_j(x; n)\exp(- nV(x)) dx= \delta_{jk}\quad\forall j,k\ge 0.$$ The authors derive Plancherel-Rotach asymptotics for $p_n(z; n)$ as $n\to\infty$, valid in every region of the plane. The precision of the asymptotics on and off the real line is remarkable. The proofs involve the Fokas-Its-Kitaev identify for the orthonormal polynomials as solutions of a Riemann-Hilbert problem, followed by the Deift-Zhou steepest descent technique. The details are intricate, but are clearly presented, and the main ideas are summarized to guide the reader through the proofs.\par As an application, the authors prove universality limits that arise in random matrix theory. These involve the weighted reproducing kernel $$K_n(x,y)= e^{-{n\over 2}(V(x)+ V(y))} \sum^{n- 1}_{j= 0} p_j(x; n)p_j(y; n)$$ and have the form $${1\over n\psi(a)} K_n\Biggl(a+ {s\over n\psi(a)}, a+{t\over n\psi(a)}\Biggr)= {\sin\pi(s- t)\over \pi(s- t)}+ O\Biggl({1\over n}\Biggr),$$ uniformly for $s$, $t$ in compact subsets of $\bbfR$. Here $a$ lies in a subinterval of the support of the equilibrium measure $\mu_V$ associated with the field $V$, and $\psi$ is the density of that equilibrium measure.
[D.S.Lubinsky (Wits)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials
15A52 Random matrices
41A60 Asymptotic problems in approximation
82B41 Random walks, etc. (statistical mechanics)

Keywords: orthogonal polynomials; universality relations; Deift-Zhou steepest descent method; Plancherel-Rotach asymptotics; random matrix theory

Cited in: Zbl 1158.15021 Zbl 1157.42007 Zbl 1112.82022 Zbl 1098.15018 Zbl 1129.42010 Zbl 1061.30035 Zbl 1035.82020 Zbl 0997.47033

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]