Law of large numbers for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation

Abstract:

We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behaviour of the flow $\phi _n$ through a discrete version $\Omega _n$ of $\Omega$ between the corresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, $\phi _n$ converges almost surely towards a constant $\phi _{\Omega }$, which is the solution of a continuous non-random min-cut problem. Moreover, we give a necessary and sufficient condition on the law of the capacity of the edges to ensure that $\phi _{\Omega } >0$.