# Upper large deviations for maximal flows through a tilted cylinder

ESAIM: Probability and Statistics (2014)

- Volume: 18, page 117-129
- ISSN: 1292-8100

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topTheret, Marie. "Upper large deviations for maximal flows through a tilted cylinder." ESAIM: Probability and Statistics 18 (2014): 117-129. <http://eudml.org/doc/273637>.

@article{Theret2014,

abstract = {We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.},

author = {Theret, Marie},

journal = {ESAIM: Probability and Statistics},

keywords = {first passage percolation; maximal flow; large deviations; tilted cylinder},

language = {eng},

pages = {117-129},

publisher = {EDP-Sciences},

title = {Upper large deviations for maximal flows through a tilted cylinder},

url = {http://eudml.org/doc/273637},

volume = {18},

year = {2014},

}

TY - JOUR

AU - Theret, Marie

TI - Upper large deviations for maximal flows through a tilted cylinder

JO - ESAIM: Probability and Statistics

PY - 2014

PB - EDP-Sciences

VL - 18

SP - 117

EP - 129

AB - We consider the standard first passage percolation model in ℤd for d ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to n and whose height is h(n) for a certain height function h. We denote this maximal flow by τn (respectively φn). We emphasize the fact that the cylinder may be tilted. We look at the probability that these flows, rescaled by the surface of the basis of the cylinder, are greater than ν(v) + ε for some positive ε, where ν(v) is the almost sure limit of the rescaled variable τn when n goes to infinity. On one hand, we prove that the speed of decay of this probability in the case of the variable τn depends on the tail of the distribution of the capacities of the edges: it can decay exponentially fast with nd−1, or with nd−1min(n,h(n)), or at an intermediate regime. On the other hand, we prove that this probability in the case of the variable φn decays exponentially fast with the volume of the cylinder as soon as the law of the capacity of the edges admits one exponential moment; the importance of this result is however limited by the fact that ν(v) is not in general the almost sure limit of the rescaled maximal flow φn, but it is the case at least when the height h(n) of the cylinder is negligible compared to n.

LA - eng

KW - first passage percolation; maximal flow; large deviations; tilted cylinder

UR - http://eudml.org/doc/273637

ER -

## References

top- [1] B. Bollobás, Graph theory, Graduate Texts in Mathematics. Springer-Verlag, New York 63 (1979). Zbl0411.05032MR536131
- [2] 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
- [3] 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
- [4] M. Théret, Upper large deviations for the maximal flow in first-passage percolation. Stochastic Process. Appl.117 (2007) 1208–1233. Zbl1121.60102MR2343936
- [5] Y. Zhang, Critical behavior for maximal flows on the cubic lattice. J. Stat. Phys.98 (2000) 799–811. Zbl0991.82019MR1749233
- [6] Y. Zhang, Limit theorems for maximum flows on a lattice. Available from arxiv.org/abs/0710.4589 (2007).

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