We consider the standard first passage percolation model in ℤ for ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to and whose height is () for a certain height function . We denote this maximal flow by
(respectively
). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows,...
We consider the standard first passage percolation on : with each edge of the lattice we associate a random capacity. We are interested in the maximal flow through a cylinder in this graph. Under some assumptions Kesten proved in 1987 a law of large numbers for the rescaled flow. Chayes and Chayes established that the large deviations far away below its typical value are of surface order, at least for the Bernoulli percolation and cylinders of certain height. Thanks to another approach we extend...
Equip the edges of the lattice ℤ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in ℝ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten [9] obtained for boxes of particular orientation.
We consider the standard first passage percolation model in ℤ for ≥2. We are interested in two quantities, the maximal flow between the lower half and the upper half of the box, and the maximal flow between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for in rational directions. Kesten and Zhang have proved the law of large numbers for and when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly...
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