Critical Behavior for Maximal Flows on the Cubic Lattice

Abstract

Let F ₀ and F _m be the top and bottom faces of the box [0, k]×[0, l]×[0, m] in Z ³. 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 Φ _k, l, m the maximal flow from F ₀ to F _m in the box. Let p _c denote the critical value for bond percolation on Z ³. It is known that Φ _k, l, m is asymptotically proportional to the area of F ₀ as m, k, l→∞, when the probability that t(e)>0 exceeds p _c , but is of lower order if the probability is strictly less than p _c . Here we consider the critical case where the probability that t(e)>0 is exactly equal to p _c , and prove that

$$\mathop {{\text{lim}}}\limits_{k,l,m \to \infty } \frac{1}{{kl}}\Phi _{k,l,m} = 0{\text{ a}}{\text{.s and in }}L_1 $$

The limiting behavior of related to surfaces on Z ³ are also considered in this paper.