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 $d\ge 2$, 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 ${\varphi }_n$ through a discrete version ${\Omega }_n$ of $\Omega$ between the corresponding discrete sets ${\Gamma }_n^1$ and ${\Gamma }_n^2$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, ${\varphi }_n$ converges almost surely towards a constant ${\varphi }_{\Omega }$, 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 ${\varphi }_{\Omega }>0$.

41 pages, 8 figures