Capacitive flows on a 2D random net

Mathematics – Probability

Scientific paper

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20 pages, 1 figure published in The Annals of Applied Probability http://www.imstat.org/aap/

Scientific paper

10.1214/08-AAP556

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.

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