Central limit theorem for first-passage percolation time across thin cylinders

• 成果类型: Article
• 署名作者: Chatterjee, Sourav; Dey, Partha S.
• 署名单位: New York University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0438-z
• 发表日期: 2013
• 页码: 613-663
• 关键词: passage percolation; shape fluctuations; UNIVERSALITY; inequalities; GROWTH

摘要

We prove that first-passage percolation times across thin cylinders of the form [0, n] x [-h (n) , h (n) ] (d-1) obey Gaussian central limit theorems as long as h (n) grows slower than n (1/(d+1)). It is an open question as to what is the fastest that h (n) can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of (1, 0, . . . , 0), we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.