Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 70-104.

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.

DOI : 10.1051/ps/2011109
Classification : 60K35, 82B43
Mots clés : first passage percolation, maximal flow, large deviation principle 
@article{PS_2013__17__70_0,
     author = {Rossignol, Rapha\"el and Th\'eret, Marie},
     title = {Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation},
     journal = {ESAIM: Probability and Statistics},
     pages = {70--104},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011109},
     mrnumber = {3007160},
     zbl = {1290.60106},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011109/}
}
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
SP  - 70
EP  - 104
VL  - 17
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011109/
DO  - 10.1051/ps/2011109
LA  - en
ID  - PS_2013__17__70_0
ER  - 
%0 Journal Article
%A Rossignol, Raphaël
%A Théret, Marie
%T Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation
%J ESAIM: Probability and Statistics
%D 2013
%P 70-104
%V 17
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2011109/
%R 10.1051/ps/2011109
%G en
%F PS_2013__17__70_0
Rossignol, Raphaël; Théret, Marie. Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 70-104. doi : 10.1051/ps/2011109. http://archive.numdam.org/articles/10.1051/ps/2011109/

[1] D. Boivin, Ergodic theorems for surfaces with minimal random weights. Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 567-599. | Numdam | MR | Zbl

[2] B. Bollobás, Graph theory. An introductory course, edited by Springer-Verlag, New York. Graduate Texts in Mathematics 63 (1979). | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. Fields 150 (2011) 635-661 | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[8] O. Garet, Capacitive flows on a 2d random net. Ann. Appl. Probab. 19 (2009) 641-660. | MR | Zbl

[9] G. Grimmett and H. Kesten, First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 (1984) 335-366. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[15] M. Théret, Upper large deviations for the maximal flow in first-passage percolation. Stoc. Proc. Appl. 117 (2007) 1208-1233. | MR | Zbl

[16] M. Théret, On the small maximal flows in first passage percolation. Ann. Fac. Sci. Toulouse 17 (2008) 207-219. | EuDML | Numdam | MR | Zbl

[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). | EuDML | Numdam | MR

[19] Y. Zhang, Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys. 98 (2000) 799-811. | MR | Zbl

[20] Y. Zhang, Limit theorems for maximum flows on a lattice. Available from arxiv.org/abs/0710.4589 (2007).

Cité par Sources :