Upper large deviations for the maximal flow through a domain of ℝd in first passage percolation

Abstract

We consider the standard first passage percolation model in the rescaled graph ℤnd / n for d ≥ 2 and a domain Ω of boundary Γ in ℝd. Let Γ1 and Γ2 be two disjoint open subsets of Γ representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behavior of the flow ϕn through a discrete version Ωn of Ω between the corresponding discrete sets Γ1 and Γn2. 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 ϕn / nd−1 above a certain constant are of volume order, that is, decays exponentially fast with nd. This article is part of a larger project in which the authors prove that this constant is the a.s. limit of ϕn / nd−1.