On the small maximal flows in first passage percolation

Marie Théret[1]

  • [1] Laboratoire de Mathémathiques d’Orsay, Université de Paris-Sud 11, bâtiment 425, 91405 Orsay cedex, France

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 1, page 207-219
  • ISSN: 0240-2963

Abstract

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We consider the standard first passage percolation on 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 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 here their result to higher cylinders, and we transport this result to the model of first passage percolation.

How to cite

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Théret, Marie. "On the small maximal flows in first passage percolation." Annales de la faculté des sciences de Toulouse Mathématiques 17.1 (2008): 207-219. <http://eudml.org/doc/10078>.

@article{Théret2008,
abstract = {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 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 here their result to higher cylinders, and we transport this result to the model of first passage percolation.},
affiliation = {Laboratoire de Mathémathiques d’Orsay, Université de Paris-Sud 11, bâtiment 425, 91405 Orsay cedex, France},
author = {Théret, Marie},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {1},
pages = {207-219},
publisher = {Université Paul Sabatier, Toulouse},
title = {On the small maximal flows in first passage percolation},
url = {http://eudml.org/doc/10078},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Théret, Marie
TI - On the small maximal flows in first passage percolation
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 1
SP - 207
EP - 219
AB - 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 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 here their result to higher cylinders, and we transport this result to the model of first passage percolation.
LA - eng
UR - http://eudml.org/doc/10078
ER -

References

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  2. Bollobás (B.).— Graph theory, volume 63 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1979. An introductory course. Zbl0411.05032MR536131
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  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. Zbl0617.60099MR847132
  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. Zbl0602.60098
  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. Zbl0591.60096MR869483
  7. Liggett (T. M.), Schonmann (R. H.), and Stacey (A. M.).— Domination by product measures. The Annals of Probability, 25(1):71-95, 1997. Zbl0882.60046MR1428500
  8. Pisztora (A.).— Surface order large deviations for Ising, Potts and percolation models. Probability Theory and Related Fields, 104(4):427-466, 1996. Zbl0842.60022MR1384040
  9. Zhang (Y.).— Critical behavior for maximal flows on the cubic lattice. Journal of Statistical Physics, 98(3-4):799-811, 2000. Zbl0991.82019MR1749233

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