# Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

Raphaël Rossignol; Marie Théret

ESAIM: Probability and Statistics (2013)

- Volume: 17, page 70-104
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topRossignol, Raphaël, and Théret, Marie. "Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation." ESAIM: Probability and Statistics 17 (2013): 70-104. <http://eudml.org/doc/273633>.

@article{Rossignol2013,

abstract = {Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in ℝ2 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.},

author = {Rossignol, Raphaël, Théret, Marie},

journal = {ESAIM: Probability and Statistics},

keywords = {first passage percolation; maximal flow; large deviation principle},

language = {eng},

pages = {70-104},

publisher = {EDP-Sciences},

title = {Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation},

url = {http://eudml.org/doc/273633},

volume = {17},

year = {2013},

}

TY - JOUR

AU - Rossignol, Raphaël

AU - Théret, Marie

TI - Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

JO - ESAIM: Probability and Statistics

PY - 2013

PB - EDP-Sciences

VL - 17

SP - 70

EP - 104

AB - Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in ℝ2 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.

LA - eng

KW - first passage percolation; maximal flow; large deviation principle

UR - http://eudml.org/doc/273633

ER -

## References

top- [1] D. Boivin, Ergodic theorems for surfaces with minimal random weights. Ann. Inst. Henri Poincaré Probab. Stat.34 (1998) 567–599. Zbl0910.60078MR1641662
- [2] B. Bollobás, Graph theory. An introductory course, edited by Springer-Verlag, New York. Graduate Texts in Mathematics 63 (1979). Zbl0411.05032MR536131
- [3] R. Cerf, The Wulff crystal in Ising and percolation models, in École d’Été de Probabilités de Saint Flour, edited by Springer-Verlag. Lect. Notes Math. 1878 (2006). Zbl1103.82010MR2241754
- [4] R. Cerf and M. Théret, 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. Zbl1228.60107MR2775823
- [5] R. Cerf and M. Théret, Lower large deviations for the maximal flow through a domain of Rd in first passage percolation. Probab. Theory Relat. Fields150 (2011) 635–661 Zbl1230.60101MR2824869
- [6] 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). Zbl1261.60089MR2895410
- [7] J. T. Chayes and L. Chayes, Bulk transport properties and exponent inequalities for random resistor and flow networks. Commun. Math. Phys.105 (1986) 133–152. Zbl0617.60099MR847132
- [8] O. Garet, Capacitive flows on a 2d random net. Ann. Appl. Probab.19 (2009) 641–660. Zbl1166.60337MR2521883
- [9] G. Grimmett and H. Kesten, First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete66 (1984) 335–366. Zbl0525.60098MR751574
- [10] 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. Zbl0143.40402MR198576
- [11] H. Kesten, 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. Zbl0602.60098MR876084
- [12] H. Kesten, Surfaces with minimal random weights and maximal flows : a higher dimensional version of first-passage percolation. Illinois J. Math.31 (1987) 99–166. Zbl0591.60096MR869483
- [13] R. Rossignol and M. Théret, Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation. Stoc. Proc. Appl.120 (2010) 873–900. Zbl1196.60168MR2610330
- [14] R. Rossignol and M. Théret, 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. Zbl1221.60144MR2744888
- [15] M. Théret, Upper large deviations for the maximal flow in first-passage percolation. Stoc. Proc. Appl.117 (2007) 1208–1233. Zbl1121.60102MR2343936
- [16] M. Théret, On the small maximal flows in first passage percolation. Ann. Fac. Sci. Toulouse17 (2008) 207–219. Zbl1152.60076MR2464099
- [17] M. Théret, Grandes déviations pour le flux maximal en percolation de premier passage. Ph.D. thesis, Université Paris Sud (2009a).
- [18] 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). MR3143735
- [19] Y. Zhang, Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys.98 (2000) 799–811. Zbl0991.82019MR1749233
- [20] Y. Zhang, Limit theorems for maximum flows on a lattice. Available from arxiv.org/abs/0710.4589 (2007).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.