A SHAPE THEOREM FOR THE ORTHANT MODEL

• 成果类型: Article
• 署名作者: Holmes, Mark; Salisbury, Thomas S.
• 署名单位: University of Melbourne; York University - Canada
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1476
• 发表日期: 2021
• 页码: 1237-1256
• 关键词: percolation

摘要

We study a particular model of a random medium, called the orthant model, in general dimensions d >= 2. Each site x is an element of Z(d) independently has arrows pointing to its positive neighbours x + e(i), i = 1, ..., d with probability p and, otherwise, to its negative neighbours x - e(i), i = 1, ..., d (with probability 1 - p). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary which is a key requirement of the subadditive ergodic theorem.