×

On the small maximal flows in first passage percolation. (English) Zbl 1152.60076

Summary: We consider the standard first passage percolation on \(\mathbb Z^{{ d }}\): 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 H. Kesten [Ill. J. Math. 31, 99–166 (1986; Zbl 0591.60096)] proved a law of large numbers for the rescaled flow. J. T. Chayes and L. Chayes [Commun. Math. Phys. 105, 133–152 (1986; Zbl 0617.60099)] 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 here their result to higher cylinders, and we transport this result to the model of first passage percolation.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] Aizenman (M.), Chayes (J. T.), Chayes (L.), Fröhlich (J.), and Russo (L.).— On a sharp transition from area law to perimeter law in a system of random surfaces. Communications in Mathematical Physics, 92:19-69, 1983. · Zbl 0529.60099
[2] Bollobás (B.).— Graph theory, volume 63 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. An introductory course. · Zbl 0411.05032
[3] Cerf (R.).— The Wulff crystal in Ising and percolation models. In École d’Été de Probabilités de Saint Flour, number 1878 in Lecture Notes in Mathematics. Springer-Verlag, 2006. · Zbl 1103.82010
[4] Chayes (J. T.) and Chayes (L.).— Bulk transport properties and exponent inequalities for random resistor and flow networks. Communications in Mathematical Physics, 105:133-152, 1986. · Zbl 0617.60099
[5] Kesten (H.).— Aspects of first passage percolation.— In École d’Été de Probabilités de Saint Flour XIV, number 1180 in Lecture Notes in Mathematics. Springer-Verlag, 1984. · Zbl 0602.60098
[6] Kesten (H.).— Surfaces with minimal random weights and maximal flows: a higher dimensional version of first-passage percolation. Illinois Journal of Mathematics, 31(1):99-166, 1987. · Zbl 0591.60096
[7] Liggett (T. M.), Schonmann (R. H.), and Stacey (A. M.).— Domination by product measures. The Annals of Probability, 25(1):71-95, 1997. · Zbl 0882.60046
[8] Pisztora (A.).— Surface order large deviations for Ising, Potts and percolation models. Probability Theory and Related Fields, 104(4):427-466, 1996. · Zbl 0842.60022
[9] Zhang (Y.).— Critical behavior for maximal flows on the cubic lattice. Journal of Statistical Physics, 98(3-4):799-811, 2000. · Zbl 0991.82019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.