Upper large deviations for maximal flows through a tilted cylinder

Abstract

We consider the standard first passage percolation model in ℤd 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 ) + e for some positive e , 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 −1 min(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 .