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Critical behavior for maximal flows on the cubic lattice. (English) Zbl 0991.82019

Summary: Let \(F_0\) and \(F_m\) be the top and bottom faces of the box \([0,k]\times [0,l]\times [0,m]\) 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 \(\Phi_{k,l,m}\) 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 \(\Phi_{k,l,m}\) is asymptotically proportional to the area of \(F_0\) as \(m,k,l\to \infty\), 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 \[ \lim_{k,l,m\to \infty} {1\over kl} \Phi_{k,l,m}= 0 \quad \text{a.s. and in }L_1. \] The limiting behavior of related to surfaces on \(Z^3\) are also considered in this paper.

MSC:

82B43 Percolation
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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