Large deviation principle for the cutsets and lower large deviation principle for the maximal flow in first passage percolation

Abstract

We consider the standard first passage percolation model in the rescaled lattice ${\mathbb{Z}}^d/n$ for $d\ge 2$ and a bounded domain $\Omega$ in ${\mathbb{R}}^d$. We denote by ${\Gamma }^1$ and ${\Gamma }^2$ two disjoint subsets of $\partial \Omega$ representing respectively the sources and the sinks, \textit{i.e.}, where the water can enter in $\Omega$ and escape from $\Omega$. A cutset is a set of edges that separates ${\Gamma }^1$ from ${\Gamma }^2$ in $\Omega$, it has a capacity given by the sum of the capacities of its edges. Under some assumptions on $\Omega$ and the distribution of the capacities of the edges, we already know a law of large numbers for the sequence of minimal cutsets $({\mathcal{E}}_n^{min}{)}_{n\ge 1}$: the sequence $({\mathcal{E}}_n^{min}{)}_{n\ge 1}$ converges almost surely to the set of solutions of a continuous deterministic problem of minimal cutset in an anisotropic network. We aim here to derive a large deviation principle for cutsets and deduce by contraction principle a lower large deviation principle for the maximal flow in $\Omega$.