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

We consider the standard first-passage percolation in ZdZd for d≥2d≥2 and we denote by ϕnd−1,h(n)ϕnd−1,h(n) the maximal flow through the cylinder ]0,n]d−1×]0,h(n)]]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, ϕnd−1,h(n)/nd−1ϕnd−1,h(n)/nd−1 converges towards a constant νν. We look now at the probability that ϕnd−1,h(n)/nd−1ϕnd−1,h(n)/nd−1 is greater than ν+εν+ε for some ε>0ε>0, and we show under some assumptions that this probability decays exponentially fast with the volume nd−1h(n)nd−1h(n) of the cylinder. Moreover, we prove a large deviation principle for the sequence (ϕnd−1,h(n)/nd−1,n∈N)(ϕnd−1,h(n)/nd−1,n∈N).