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 Z d with a distribution G on R + that admits an exponential moment. We study the maximal flow between a compact convex subset A of 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 φ(nA)/n d−1 almost surely converges towards a deterministic constant depending on A. This constant corresponds to the capacity of the boundary ∂A of A and is the integral of a deterministic function over ∂A. This result was shown in dimension 2 and conjectured for higher dimensions by Garet in [6].