Discrete orthogonal polynomial ensembles and the Plancherel measure
type: Item
instance of: scholarly article
title: Discrete orthogonal polynomial ensembles and the Plancherel measure
zbMATH Open document ID: 0984.15020
DOI: 10.2307/2661375
author: Kurt Johansson
published in: Annals of Mathematics. Second Series
publication date: 2002-05-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9906120
full work available at URL: https://eudml.org/doc/121768
reviewed by: Yueh-er Kuo
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 1625953
zbMATH Keywords: orthogonal polynomial ensembles
zbMATH Keywords: probability measures
zbMATH Keywords: Meixner ensemble
zbMATH Keywords: Charlier ensembles
zbMATH Keywords: Krawtchouk ensemble
zbMATH Keywords: random matrix
zbMATH Keywords: Gaussian unitary ensemble
MaRDI profile type: Publication
OpenAlex ID: W2118974146
arXiv ID: math/9906120
review text: The author considers orthogonal polynomial ensembles. The Meixner ensemble is related to a two-dimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zig-zag paths in random domino tilings of the Aztec diamond.NEWLINENEWLINENEWLINEThe author uses the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of \textit{C. A. Tracy} and \textit{H. Widom} [J. Stat. Phys. 92, No. 5-6, 809-835 (1998; Zbl 0942.60099)]. The author obtains the Plancherel measure on partitions, and using this the author proves a conjecture of \textit{J. Baik}, \textit{P. Deift} and \textit{K. Johansson} [Geom. Funct. Anal. 10, No. 4, 702-731 (2000; Zbl 0963.05133)] that under the Plancherel measure, the distributions of the lengths of the first \(k\) rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the \(k\) largest eigenvalues of a random matrix from the Gaussian unitary ensemble. In this problem a certain discrete kernel plays an important role.
description: scientific article; zbMATH DE number 1625953