The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights

By K. Mclaughlin, Peter D. Miller. 2008

169 citations recorded by the source corpus.

We obtain Plancherel-Rotach-type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form on the real line, assuming that V has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application, we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. We develop a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/ecb48b538ac6467d0f81187dbe4acffc1a8aafc4

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Citations in the corpus

A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, methodology): “The method works best in the class of analytic weights, but progress has recently been made for weights that have only a finite number of derivatives [91].”
Long time asymptotic behavior of the focusing nonlinear Schrödinger equation (2016, unknown): “d [9] to E(x,t) = O t−(1/2+κ) for any 0 <κ<1/4 under the much weaker assumption that ψ0 belonged to the weighted Sobolev space H1,1 = f∈ L2(R) : xf, f′ ∈ L2(R) . Recently, McLaughlin and Miller [15, 16], developed a method of asymptotic analysis of Riemann-Hilbert problems based on ∂problems, ”
On the Asymptotic Stability of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}$$\end{document}-Soliton Solutions (2016, methodology): “Our proof uses the ∂ method for contour deformation introduced in McLaughlin–Miller [37,38] and Dieng–McLaughlin [21], which allows us to consider initial data with only a small amount of regularity while simultaneously simplifying many of the necessary estimates of more standard steepest descent.”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, methodology): “The recently developed tools of asymptotic analysis of the RH problems of a certain class, such as the non-linear steepest descent method of Deift and Zhou (see [22], [23], [24], as well as [10]) combined with the ∂-problem ([62], [63]), have become powerful weapons in the study of the strong asympt”