### Threshold phenomena on product spaces: BKKKL revisited (once more).

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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 provide a new exponential concentration inequality for first passage percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm ( (2003)) which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol ( (2006)), Falik and Samorodnitsky ( (2007)).

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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