Size of a minimal cutset in supercritical first passage percolation

Abstract

We consider the standard model of i.i.d. first passage percolation on Z^d given a distribution G on [0, +$\infty$] (including +$\infty$). We suppose that G({0}) > 1 -- p\_c(d), i.e., the edges of positive passage time are in the subcritical regime of percolation on Z^d. We consider a cylinder of basis an hyperrectangle of dimension d -- 1 whose sides have length n and of height h(n) with h(n) negligible compared to n (i.e., h(n)/n $\rightarrow$ 0 when n goes to infinity). We study the maximal flow from the top to the bottom of this cylinder. We already know that the maximal flow renormalized by n^(d--1) converges towards the flow constant which is null in the case G({0}) > 1 -- p\_c (d). The study of maximal flow is associated with the study of sets of edges of minimal capacity that cut the top from the bottom of the cylinder. If we denote by $\psi$\_n the minimal cardinal of such a set of edges, we prove here that $\psi$\_n /n^(d--1) converges almost surely towards a constant.