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Capacitive flows on a 2D random net. (English) Zbl 1166.60337

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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