2009/07/03 by Marie Théret, Theret, Marie
Mathematics · Physics and Astronomy · #60F10. #60K35 #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.0907.0614
openalex publication_date 2009/07/03 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
We consider the standard first passage percolation model in \ZZd 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 τn (respectively ϕn). 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)+\eps for some positive \eps, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decays exponentially fast with nd-1, or with nd-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 ϕn 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 ϕn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.