AbstractLet 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 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 Z3 are also considered in this paper.
[1]
U. Krengel,et al.
Ergodic theorems for superadditive processes.
,
1981
.
[2]
Harry Kesten,et al.
First-passage percolation, network flows and electrical resistances
,
1984
.
[3]
J. Chayes,et al.
Bulk transport properties and exponent inequalities for random resistor and flow networks
,
1986
.
[4]
H. Kesten.
Surfaces with minimal random weights and maximal flows: A higher dimensional version of first-passage percolation
,
1987
.
[5]
Charles M. Newman,et al.
Percolation in half-spaces: equality of critical densities and continuity of the percolation probability
,
1991
.
[6]
Multiparameter Subadditive Processes
,
1976
.