The maximal flow from a compact convex subset to infinity in first passage percolation on $\mathbb{Z}^{d}$

Abstract

We consider the standard first passage percolation model on $\mathbb{Z}^{d}$ with a distribution $G$ on $\mathbb{R}^{+}$ that admits an exponential moment. We study the maximal flow between a compact convex subset $A$ of $\mathbb{R}^{d}$ and infinity. The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut $A$ from infinity. We prove that the rescaled maximal flow between $nA$ and infinity $\phi (nA)/n^{d-1}$ almost surely converges toward a deterministic constant depending on $A$. This constant corresponds to the capacity of the boundary $\partial A$ of $A$ and is the integral of a deterministic function over $\partial A$. This result was shown in dimension $2$ and conjectured for higher dimensions by Garet in (Annals of Applied Probability 19 (2009) 641–660).