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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 $d\geq 2$ and a bounded domain $Ω$ in $\mathbb R ^d$. We denote by $Γ^1$ and $Γ^2$ two disjoint subsets of $\partial Ω$ representing respectively the source and the sink, i.e., where the water can enter in $Ω$ and escape from $Ω$. A maximal stream is a vector measure $\overrightarrowμ_n^{max}$ that describes how the maximal amount of fluid can enter through $Γ^1$ and spreads in $Ω$. Under some assumptions on $Ω$ and $G$, we already know a law of large number for $\overrightarrowμ_n^{max}$. The sequence $(\overrightarrowμ_n^{max})_{n\geq 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 $Ω$.