Asymptotics of Orthogonal Polynomials via the Koosis Theorem

ar X iv : m at h - ph / 0 51 00 12 v 1 3 O ct 2 00 5 ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS VIA THE KOOSIS THEOREM

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed...

Semiclassical asymptotics of orthogonal polynomials , Riemann - Hilbert problem , and universality in the matrix model

We derive semiclassical asymptotics for the orthogonal polynomials P n (z) on the line with respect to the exponential weight exp(−NV (z)), where V (z) is a double-well quartic polynomial, in the limit when n, N → ∞. We assume that ε ≤ (n/N) ≤ λ cr − ε for some ε > 0, where λ cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously...

Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory

We consider polynomials orthogonal on [0,∞) with respect to Laguerre-type weights w(x) = xe, 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 coeffic...

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem

A universality theorem for ratios of random characteristic polynomials

We consider asymptotics of ratios of random characteristic polynomials associated with orthogonal polynomial ensembles. Under some natural conditions on the measure in the definition of the orthogonal polynomial ensemble we establish a universality limit for these ratios. c ⃝ 2012 Elsevier Inc. All rights reserved.