A growth model in a random environment

2000/11/30 by Janko Gravner, Craig A. Tracy, Harold Widom
Mathematics · Physics and Astronomy · #math.PR #cond-mat.dis-nn #math.CO #msc:60K35 #msc:05A16 #msc:33E17 #msc:82B44

paper · pdf

published as Ann. Probab. 30 (2002), 1340-1368. · 31 pages, 5 figures

arxiv created 2001/07/02 · arxiv updated 2009/11/30

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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