Shape fluctuations are different in different directions

• 成果类型: Article
• 署名作者: Zhang, Yu
• 署名单位: University of Colorado System; University of Colorado at Colorado Springs
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000213
• 发表日期: 2008
• 页码: 331-362
• 关键词: 1st passage percolation; 2 dimensions; divergence

摘要

We consider the first passage percolation model on Z(2). In this model, we assign independently to each edge e a passage time t(e) with a common distribution F. Let T(u, v) be the passage time from u to v. In this paper, we show that, whenever F(0) < p(c), sigma(2)(T((0, 0), (n, 0))) >= C log n for all n >= 1. Note that if F satisfies an additional special condition, inf supp(F) = r > 0 and F(r) > (p) over right arrow (c), it is known that there exists M such that for all n, sigma(2)(T((0, 0), (n, n))) <= M. These results tell us that shape fluctuations not only depend on distribution F, but also on direction. When showing this result, we find the following interesting geometrical property. With the special distribution above, any long piece with r-edges in an optimal path from (0, 0) to (n, 0) has to be very circuitous.