Upper large deviations for maximal flows through a tilted cylinder

We consider the standard first passage percolation model in $\ZZ^d$ for $d\geq 2$ 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 $\tau_n$ (respectively $\phi_n$). 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 $\nu(\vec{v})+\eps$ for some positive $\eps$, where $\nu(\vec{v})$ is the almost sure limit of the rescaled variable $\tau_n$ when $n$ goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable $\tau_n$ depends on the tail of the distribution of the capacities of the edges: it can decays 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 $\phi_n$ 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 $\nu(\vec{v})$ is not in general the almost sure limit of the rescaled maximal flow $\phi_n$, but it is the case at least when the height $h(n)$ of the cylinder is negligible compared to $n$.