The maximal flow from a compact convex subset to infinity in first passage percolation on 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 $\ddot{I}•$(nA)/n^ (d--1) almost surely converges towards 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 [6].