Lower large deviations for the maximal flow through a domain of $${\mathbb{R}^d}$$ in first passage percolation

Abstract

We consider the standard first passage percolation model in the rescaled graph \({\mathbb{Z}^d/n}\) for d ≥ 2, and a domain Ω of boundary Γ in \({\mathbb{R}^d}\) . 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 behaviour of the flow \({\phi_n}\) through a discrete version Ωn of Ω 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 lower large deviations of \({\phi_n/ n^{d-1}}\) below a certain constant are of surface order.