Upper large deviations for maximal flows through a tilted cylinder. (28th November 2013)

• Record Type: Journal Article
• Authors: Theret, Marie
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 117
• Page End: 129
• Publication Date: 2013-11-28
• Subjects: First passage percolation, -- maximal flow, -- large deviations; Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2012029
• Languages: English
• ISSNs: 1292-8100

Abstract

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 −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 .