Law of large numbers for the maximal flow through a domain of $\mathbb{R}^d$ in first passage percolation
We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^d/n$ for $d\geq 2$, and a domain $Ω$ of boundary $Γ$ in $\mathbb{R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$, representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behaviour of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, $ϕ_n$ converges almost surely towards a constant $ϕ_Ω$, 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 $ϕ_Ω >0$.