Maximal stream and minimal cutset for first passage percolation through a domain of $\mathbb{R}^{d}$

Abstract

We consider the standard first passage percolation model in the rescaled graph $\mathbb{Z}^{d}/n$ for $d\geq2$ 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 $. A law of large numbers for the maximal flow from $\Gamma ^{1}$ to $\Gamma ^{2}$ in $\Omega $ is already known. In this paper we investigate the asymptotic behavior of a maximal stream and a minimal cutset. A maximal stream is a vector measure $\vec{\mu}_{n}^{\max}$ that describes how the maximal amount of fluid can cross $\Omega $. Under conditions on the regularity of the domain and on the law of the capacities of the edges, we prove that the sequence $(\vec{\mu}_{n}^{\max})_{n\geq1}$ converges a.s. to the set of the solutions of a continuous deterministic problem of maximal stream in an anisotropic network. A minimal cutset can been seen as the boundary of a set $E_{n}^{\min}$ that separates $\Gamma ^{1}$ from $\Gamma ^{2}$ in $\Omega $ and whose random capacity is minimal. Under the same conditions, we prove that the sequence $(E_{n}^{\min})_{n\geq1}$ converges toward the set of the solutions of a continuous deterministic problem of minimal cutset. We deduce from this a continuous deterministic max-flow min-cut theorem and a new proof of the law of large numbers for the maximal flow. This proof is more natural than the existing one, since it relies on the study of maximal streams and minimal cutsets, which are the pertinent objects to look at.