Capacitive flows on a 2D random net

This paper concerns maximal flows on $\mathbb{Z}^2$ traveling from a convex set to infinity, the flows being restricted by a random capacity. For every compact convex set $A$, we prove that the maximal flow $\Phi(nA)$ between $nA$ and infinity is such that $\Phi(nA)/n$ almost surely converges to the integral of a deterministic function over the boundary of $A$. The limit can also be interpreted as the optimum of a deterministic continuous max-flow problem. We derive some properties of the infinite cluster in supercritical Bernoulli percolation.