zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. (English) Zbl 0956.42014
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. 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. 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.

42C05General theory of orthogonal functions and polynomials
33C05Classical hypergeometric functions, ${}_2F_1$
15B52Random matrices
33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI Link EuDML arXiv