Large deviations for the chemical distance in supercritical Bernoulli percolation

• 成果类型: Article
• 署名作者: Garet, Olivier; Marchand, Regine
• 署名单位: Universite de Orleans; Universite de Lorraine
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000881
• 发表日期: 2007
• 页码: 833-866
• 关键词: 1st-passage percolation; phase

摘要

The chemical distance D(x, y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behavior of this random metric, and we prove that, for an appropriate norm mu depending on the dimension and the percolation parameter, the probability of the event {0 <-> x, D(0, x)/mu(x) is not an element of (1 - epsilon, 1 + epsilon} exponentially decreases when vertical bar vertical bar x vertical bar vertical bar(1) tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.