LONG-RANGE LAST-PASSAGE PERCOLATION ON THE LINE

• 成果类型: Article
• 署名作者: Foss, Sergey; Martin, James B.; Schmidt, Philipp
• 署名单位: Heriot Watt University; Russian Academy of Sciences; University of Oxford
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP920
• 发表日期: 2014
• 页码: 198-234
• 关键词: graphs

摘要

We consider directed last-passage percolation on the random graph G = (V, E) where V = Z and each edge (i, j), for i < j is an element of Z, is present in E independently with some probability p is an element of (0, 1]. To every (i, j) is an element of E we attach i.i.d. random weights v(i, j) > 0. We are interested in the behaviour of w(0,n), which is the maximum weight of all directed paths from 0 to n, as n -> infinity. We see two very different types of behaviour, depending on whether E[v(i,j)(2)] < infinity or E[v(i,j)(2)] = infinity. In the case where E[v(i,j)(2)] < infinity we show that the process has a certain regenerative structure, and prove a strong law of large numbers and, under an extra assumption, a functional central limit theorem. In the situation where E[v(i,j)(2)] = infinity we obtain scaling laws and asymptotic distributions expressed in terms of a continuous last-passage percolation model on [0,1]; these are related to corresponding results for two-dimensional lastpassage percolation with heavy-tailed weights obtained in Hambly and Martin