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

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