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 ℤ_n^d / n for d ≥ 2 and a domain Ω of boundary Γ in ℝ^d. 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 ϕ_n through a discrete version Ω_n of Ω between the corresponding discrete sets Γ¹ and Γ_n². 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 ϕ_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 ϕ_n / n^d−1.