Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Details Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

2009-07-29

arxiv.org/abs/0907.5112v2

Mathematics

Probability

27 pages, 4 figures; minor modifications

Scientific paper

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained by Grimmett and Kesten (1984) for rectangles of particular orientation.

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