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Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation. (English) Zbl 1221.60144

The paper deals with the standard first passage percolation model in the finite-dimensional integer lattice. The authors 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. The probabilities that these quantities rescaled to the surface of the basis of the box are abnormally small are studied. For the first quantity, the box can have any orientation, whereas for the second quantity, it is required either that the box is sufficiently flat or that its sides are parallel to the coordinate hyperplanes. The authors show that these probabilities decay exponentially fast with the surface of the basis of the box, when it grows to infinity. In addition, they prove an associated large deviation principle for the scaled quantities and improve the conditions required to obtain the law of large numbers for these quantities.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
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References:

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