Limit theorems for height fluctuations in a class of discrete space and time growth models
type: Item
instance of: scholarly article
title: Limit theorems for height fluctuations in a class of discrete space and time growth models
zbMATH Open document ID: 0989.82030
DOI: 10.1023/A:1004879725949
author: Janko Gravner
author: Craig A. Tracy
author: Harold Widom
published in: Journal of Statistical Physics
publication date: 2001-08-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0005133
review text: We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function \(h_t(x)\) with corner initialization. We prove, with one exception, that the limiting distribution function of \(h_t(x)\) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large \(x\) and large \(t\) the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual Robinson-Schensted-Knuth algorithm, Gessel's theorem, the Borodin-Okounkov identity and a novel, rigorous saddle point analysis. In the fixed \(x\), large \(t\) regime, we find a Brownian motion representation. The model is equivalent to the Seppäläinen-Johansson model. Hence some of our results are not new, but the proofs are.
Mathematics Subject Classification ID: 82C41
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 1638192
zbMATH Keywords: growth processes
zbMATH Keywords: shape fluctuations
zbMATH Keywords: limit theorems
zbMATH Keywords: Airy kernel
zbMATH Keywords: invariance principle
zbMATH Keywords: limiting distribution function
zbMATH Keywords: Fredholm determinant
zbMATH Keywords: saddle point analysis
MaRDI profile type: Publication
arXiv ID: math/0005133
OpenAlex ID: W1821888604
Wikidata QID: Q56950649
description: scientific article; zbMATH DE number 1638192