Upper large deviations for maximal flows through a tilted cylinder

D.M.A., E.N.S., 45 rue d’Ulm, 75230 Paris Cedex 05, France
marie.theret@ens.fr

Received: 3 July 2009

Abstract

We consider the standard first passage percolation model in ℤ^d for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τ_n (respectively φ_n). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τ_n when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τ_n depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with n^d−1, or with n^d−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φ_n decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φ_n, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.