The maximal flow from a compact convex subset to infinity in first passage percolation on Zd

We consider the standard first passage percolation model on Zd with a distribution G on R+ that admits an exponential moment. We study the maximal flow between a compact convex subset A of Rd 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 .(nA)/nd.1 almost surely converges toward 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 (Annals of Applied Probability 19 (2009) 641.660).