We consider the standard first passage percolation model in the rescaled lattice \( Z^d/n\) for \( d>= 2\) : with each edge e we associate a random capacity \( c(e)>= 0\) such that the family \( (c(e))_e\) is independent and identically distributed with a common law G. We interpret this capacity as a rate of flow, i.e., it corresponds to the maximal amount of water that can cross the edge per unit of time. We consider a bounded connected domain \( \Omega\) in \(R^d\) and two disjoint subsets of the boundary of \( \Omega\) representing respectively the source and the sink, i.e., where the water can enter in \( \Omega\) and escape from \( \Omega\). We are interested in the maximal flow, i.e., the maximal amount of water that can enters through \( \Omega\) per unit of time. A stream is a function on the edges that describes how the water circulates in $\Omega$. In this talk, we will present a large deviation principle for streams and deduce by contraction principle an upper large deviation principle for maximal flow in $\Omega$.
This is a joint work with Marie Théret.