MAXIMAL STREAM AND MINIMAL CUTSET FOR FIRST PASSAGE PERCOLATION THROUGH A DOMAIN OF Rd

• 成果类型: Article
• 署名作者: Cerf, Raphael; Theret, Marie
• 署名单位: Universite Paris Saclay; Universite Paris Saclay; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP851
• 发表日期: 2014
• 页码: 1054-1120
• 关键词: large deviations; FLOW

摘要

We consider the standard first passage percolation model in the resealed graph Z(d)/n for d >= 2 and a domain Omega of boundary Gamma in R-d Let Gamma(1) and Gamma(2) be two disjoint open subsets of Gamma, representing the parts of Gamma through which some water can enter and escape from Omega. A law of large numbers for the maximal flow from Gamma(1), to Gamma(2) in Omega is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure (mu) over right arrow (max)(n) that describes how the maximal amount of fluid can cross Omega. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence ((mu) over right arrow (max)(n) )(n >=)1 converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set E-n(min) that separates rl from Gamma(2) in Omega and whose random capacity is minimal. Under the same conditions, we prove that the sequence (E-n(min))(n)>= 1 converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at.