zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. (English) Zbl 0944.42013

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: be real-valued and analytic on , with

lim |x| V(x)/log(1+x 2 )=·

For n1, let {p k (x;n)} k=0 denote the sequence of orthonormal polynomials for the weight exp(-nV), so that

p k (x;n)p j (x;n)exp(-nV(x))dx=δ jk j,k0·

The authors derive Plancherel-Rotach asymptotics for p n (z;n) as n, 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.

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 2(V(x)+V(y)) j=0 n-1 p j (x;n)p j (y;n)

and have the form

1 nψ(a)K n a + s nψ(a) , a + t nψ(a)=sinπ(s-t) π(s-t)+O1 n,

uniformly for s, t in compact subsets of . Here a lies in a subinterval of the support of the equilibrium measure μ V associated with the field V, and ψ is the density of that equilibrium measure.


MSC:
42C05General theory of orthogonal functions and polynomials
15A52Random matrices (MSC2000)
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
82B41Random walks, random surfaces, lattice animals, etc. (statistical mechanics)