Upper large deviations for the maximal flow in first-passage percolation

We consider the standard first-passage percolation in Z d for d ≥ 2 and we denote by ϕ n d − 1 , h ( n ) the maximal flow through the cylinder ] 0 , n ] d − 1 × ] 0 , h ( n ) ] from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension 3: under some assumptions, ϕ n d − 1 , h ( n ) / n d − 1 converges towards a constant ν . We look now at the probability that ϕ n d − 1 , h ( n ) / n d − 1 is greater than ν + ε for some ε > 0 , and we show under some assumptions that this probability decays exponentially fast with the volume n d − 1 h ( n ) of the cylinder. Moreover, we prove a large deviation principle for the sequence ( ϕ n d − 1 , h ( n ) / n d − 1 , n ∈ N ) .