UPPER LARGE DEVIATIONS FOR THE MAXIMAL FLOW THROUGH A DOMAIN OF R-d IN FIRST PASSAGE PERCOLATION

作者：Cerf Raphael^*; Theret Marie

来源：Annals of Applied Probability, 2011, 21(6): 2075-2108.

DOI：10.1214/10-AAP732

• First passage percolation
• maximal flow
• minimal cut
• large deviations

We consider the standard first passage percolation model in the rescaled 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. We investigate the asymptotic behavior of the flow phi(n) through a discrete version Omega(n) of Omega between the corresponding discrete sets Gamma(1)(n) and Gamma(2)(n). We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviations of phi(n)/n(d-1) above a certain constant are of volume order, that is, decays exponentially fast with n(d). This article is part of a larger project in which the authors prove that this constant is the a.s. limit of phi(n)/n(d-1).

• 出版日期2011-12