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

TL;DR

This paper proves a law of large numbers for the maximal flow in a first passage percolation model in $bR^d$, showing almost sure convergence to a deterministic value and characterizing when this limit is positive.

Contribution

It establishes almost sure convergence of the maximal flow to a deterministic limit and provides conditions on edge capacities for the flow to be positive.

Findings

Flow converges almost surely to a deterministic constant.

Characterization of when the limiting flow is positive.

Conditions on capacity law ensuring positive flow.

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 $\phi_{\Omega} >0$.