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July 2002 A growth model in a random environment
Janko Gravner, Craig A. Tracy, Harold Widom
Ann. Probab. 30(3): 1340-1368 (July 2002). DOI: 10.1214/aop/1029867130

Abstract

We consider a model of interface growth in two dimensions, given by a height function on the sites of the one-dimensional integer lattice. According to the discrete time update rule, the height above the site x increases to the height above $x-1$, if the latter height is larger; otherwise the height above x increases by 1 with probability $p_x$. We assume that $p_x$ are chosen independently at random with a common distribution F and that the initial state is such that the origin is far above the other sites. We explicitly identify the asymptotic shape and prove that, in the pure regime, the fluctuations about that shape, normalized by the square root of time, are asymptotically normal. This contrasts with the quenched version: conditioned on the environment, and normalized by the cube root of time, the fluctuations almost surely approach a distribution known from random matrix theory.

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Janko Gravner. Craig A. Tracy. Harold Widom. "A growth model in a random environment." Ann. Probab. 30 (3) 1340 - 1368, July 2002. https://doi.org/10.1214/aop/1029867130

Information

Published: July 2002
First available in Project Euclid: 20 August 2002

zbMATH: 1021.60083
MathSciNet: MR1920110
Digital Object Identifier: 10.1214/aop/1029867130

Subjects:
Primary: 60K35
Secondary: 05A16 , 33E17 , 82B44

Keywords: Fluctuations , Fredholm determinant , Growth model , Painlevé II , saddle point method , Time constant

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.30 • No. 3 • July 2002
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