A growth model in a random environment

• 成果类型: Article
• 署名作者: Gravner, J; Tracy, CA; Widom, H
• 署名单位: University of California System; University of California Davis; University of California System; University of California Santa Cruz
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1340-1368
• 关键词: limiting-shape; distributions; fluctuations

摘要

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 I 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.