Large deviation principle for the streams and the maximal flow in first passage percolation

We consider the standard first passage percolation model in the rescaled lattice ${\mathbb{Z}}^d$ for $d\ge 2$ and a bounded domain $\mathrm{\Omega }$ in $\mathbb R ^d$\mathrm{.}Wedenoteby$\Gamma^1$and$\Gamma^2$twodisjointsubsetsof$\partial \Omega$ representing respectively the source and the sink, i.e., where the water can enter in $\mathrm{\Omega }$ and escape from $\mathrm{\Omega }$. A maximal stream is a vector measure ${\overrightarrow{\mu }}_n^{max}$ that describes how the maximal amount of fluid can enter through ${\mathrm{\Gamma }}^1$ and spreads in $\mathrm{\Omega }$. Under some assumptions on $\mathrm{\Omega }$ and $G$, we already know a law of large number for ${\overrightarrow{\mu }}_n^{max}$. The sequence $({\overrightarrow{\mu }}_n^{max}{)}_{n\ge 1}$ converges almost surely to the set of solutions of a continuous deterministic problem of maximal stream in an anisotropic network. We aim here to derive a large deviation principle for streams and deduce by contraction principle the existence of a rate function for the upper large deviations of the maximal flow in $\mathrm{\Omega }$.