First passage percolation has sublinear distance variance

• 成果类型: Article
• 署名作者: Benjamini, I; Kalai, G; Schramm, O
• 署名单位: Weizmann Institute of Science; Hebrew University of Jerusalem; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1970-1978
• 关键词: increasing subsequences; fluctuations

摘要

Let 0 < a < b < infinity, and for each edge e of Z(d) let omega(e) = a or omega(e) = b, each with probability 1/2, independently. This induces a random metric dist(omega) on the vertices of Zd, called first passage percolation. We prove that for d > 1, the distance dist(omega)(0, v) from the origin to a vertex v, \v\ > 2, has variance bounded by C\v\/log \v\, where C = C(a, b, d) is a constant which may only depend on a, b and d. Some related variants are also discussed.