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Lower large deviations for the maximal flow through tiltedcylinders in two-dimensional first passage percolation

Published online by Cambridge University Press:  06 December 2012

Raphaël Rossignol  and
Marie Théret
Raphaël Rossignol
Affiliation:
Université Paris Sud, Laboratoire de Mathématiques, bâtiment 425, 91405 Orsay Cedex, France.. [email protected]
Marie Théret
Affiliation:
École Normale Supérieure, Département de Mathématiques et Applications, 45 rue d’Ulm, 75230 Paris Cedex 05, France. ; [email protected]
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Abstract

Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law oflarge numbers is known for the maximal flow crossing a rectangle in ℝ2 when theside lengths of the rectangle go to infinity. We prove that the lower large deviations areof 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.

Keywords

First passage percolationmaximal flowlarge deviation principle.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 70 - 104
Copyright
© EDP Sciences, SMAI, 2012

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References

Références

Boivin, D., Ergodic theorems for surfaces with minimal random weights. Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 567–599. Google Scholar
Bollobás, B., Graph theory. An introductory course, edited by Springer-Verlag, New York. Graduate Texts in Mathematics 63 (1979). Google Scholar
Cerf, R., The Wulff crystal in Ising and percolation models, in École d’É té de Probabilités de Saint Flour, edited by Springer-Verlag. Lect. Notes Math. 1878 (2006). Google Scholar
Cerf, R. and Théret, M., Law of large numbers for the maximal flow through a domain of Rd in first passage percolation. Trans. Amer. Math. Soc. 363 (2011) 3665–3702. Google Scholar
Cerf, R. and Théret, M., Lower large deviations for the maximal flow through a domain of Rd in first passage percolation. Probab. Theory Relat. Fields 150 (2011) 635–661 Google Scholar
R. Cerf and M. Théret, Upper large deviations for the maximal flow through a domain of Rd in first passage percolation. To appear in Ann. Appl. Probab., available from arxiv.org/abs/0907.5499 (2009c).
Chayes, J. T. and Chayes, L., Bulk transport properties and exponent inequalities for random resistor and flow networks. Commun. Math. Phys. 105 (1986) 133–152. Google Scholar
Garet, O., Capacitive flows on a 2d random net. Ann. Appl. Probab. 19 (2009) 641–660. Google Scholar
Grimmett, G. and Kesten, H., First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 (1984) 335–366. Google Scholar
J.M. Hammersley and D.J.A. Welsh, First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, in Proc. Internat. Res. Semin., Statist. Lab. Univ. California, Berkeley, Calif. Springer-Verlag, New York (1965) 61–110.
Kesten, H., Aspects of first passage percolation, in École d’été de probabilités de Saint-Flour, XIV–1984, edited by Springer, Berlin. Lect. Notes Math. 1180 (1986) 125–264. Google Scholar
Kesten, H., Surfaces with minimal random weights and maximal flows : a higher dimensional version of first-passage percolation. Illinois J. Math. 31 (1987) 99–166. Google Scholar
Rossignol, R. and Théret, M., Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation. Stoc. Proc. Appl. 120 (2010) 873–900. Google Scholar
Rossignol, R. and Théret, M., Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation. Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 1093–1131. Google Scholar
Théret, M., Upper large deviations for the maximal flow in first-passage percolation. Stoc. Proc. Appl. 117 (2007) 1208–1233. Google Scholar
Théret, M., On the small maximal flows in first passage percolation. Ann. Fac. Sci. Toulouse 17 (2008) 207–219. Google Scholar
M. Théret, Grandes déviations pour le flux maximal en percolation de premier passage. Ph.D. thesis, Université Paris Sud (2009a).
M. Théret, Upper large deviations for maximal flows through a tilted cylinder. To appear in ESAIM : Probab. Stat., available from arxiv.org/abs/0907.0614 (2009b).
Zhang, Y., Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys. 98 (2000) 799–811. Google Scholar
Y. Zhang, Limit theorems for maximum flows on a lattice. Available from arxiv.org/abs/0710.4589 (2007).