Critical Behavior for Maximal Flows on the Cubic Lattice

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Let F0 and Fm be the top and bottom faces of the box [0, k]×[0, l]×[0, m] in Z3. To each edge e in the box, we assign an i.i.d. nonnegative random variable t(e) representing the flow capacity of e. Denote by Φk, l, m the maximal flow from F0 to Fm in the box. Let pc denote the critical value for bond percolation on Z3. It is known that Φk, l, m is asymptotically proportional to the area of F0 as m, k, l→∞, when the probability that t(e)>0 exceeds pc, but is of lower order if the probability is strictly less than pc. Here we consider the critical case where the probability that t(e)>0 is exactly equal to pc, and prove thatThe limiting behavior of related to surfaces on Z3 are also considered in this paper.

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