Universality limits for random matrices and de Branges spaces of entire functions
type: Item
instance of: scholarly article
title: Universality limits for random matrices and de Branges spaces of entire functions
zbMATH Open document ID: 1184.46029
author: Doron S. Lubinsky
published in: Journal of Functional Analysis
publication date: 2009-06-17T00:00:00Z
review text: In the theory of orthogonal polynomials the so-called ``universality law'' has been studied from different viewpoints and with measures satisfying less and less stringent conditions. If \(\{p_k(x)\}\) are the orthonormal polynomials with respect to a finite positive Borel measure \(\mu\) with compact support \(J\) on the real line, the simplest case of this law is given in terms of the reproducing kernel \[ K_n(x,y)=\sum_{k=0}^n\,p_k(x)p_k(y),\;\tilde{K}_n(x,y)=w(x)^{1/2}w(y)^{1/2}K_n(x,y). \] Let \(w={\text{d}\mu\over\text{d}x}\) be the Radon-Nikodým derivative of \(\mu\). The paper starts (Section 1) with some notation and states a typical result in the field: Theorem 1.1. Let \(\mu\) be a finite positive Borel measure on the real line with compact support. Let \(J\subset\text{supp}(\mu)\) be compact and such that \(\mu\) is absolutely continuous at each point of \(J\). The following are equivalent: A. uniformly for \(\xi\in J\) and \(a\) in compact subsets of the real line, \[ \lim_{n\rightarrow\infty}\,{K_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{a\over\tilde{K}_n(\xi,\xi)}\right)\over K_n(\xi,\xi)}=1 \] B. uniformly for \(\xi\in J\) and \(a,b\) in compact subsets of the complex plane, \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over K_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \] In this paper, the author explores the possible limits of subsequences of the sequence \(\{f_n\}\), where \[ f_n(a,b)={K_n\left(\xi_n+{a\over\tilde{K}_n(\xi_n,\xi_n)}, \xi_n+{b\over\tilde{K}_n(\xi_n,\xi_n)}\right)\over K_n(\xi,\xi)}, \] and \(\{\xi_n\}\) is a sequence of real numbers. Since the \(\{K_n\}\) are reproducing kernels for polynomials, it is no surprise that the limits of the subsequences turn out to be reproducing kernels for suitable entire functions; here, the setting of de Branges spaces, Hermite-Biehler class and Paley-Wiener spaces turn up. Let \(E\) be an entire function without zeros …
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 5565772
zbMATH Keywords: random matrices
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: universality limits
zbMATH Keywords: de Branges spaces
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.02.021
OpenAlex ID: W1989213926
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DOI: 10.1016/J.JFA.2009.02.021
description: scientific article; zbMATH DE number 5565772