Capacitive flows on a 2D random net

Capacitive flows on a 2D random net

Olivier Garet

Source: Ann. Appl. Probab. Volume 19, Number 2 (2009), 641-660.

Abstract

This paper concerns maximal flows on ℤ² traveling from a convex set to infinity, the flows being restricted by a random capacity. For every compact convex set A, we prove that the maximal flow Φ(nA) between nA and infinity is such that Φ(nA)/n almost surely converges to the integral of a deterministic function over the boundary of A. The limit can also be interpreted as the optimum of a deterministic continuous max-flow problem. We derive some properties of the infinite cluster in supercritical Bernoulli percolation.

First Page: Show

Primary Subjects: 60K35

Secondary Subjects: 82B43

Keywords: First-passage percolation; maximal flows

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1241702245
Digital Object Identifier: doi:10.1214/08-AAP556
Zentralblatt MATH identifier: 05566095
Mathematical Reviews number (MathSciNet): MR2521883

back to Table of Contents

References

[1] Aizenman, M., Chayes, J. T., Chayes, L., Fröhlich, J. and Russo, L. (1983). On a sharp transition from area law to perimeter law in a system of random surfaces. Comm. Math. Phys. 92 19–69.

Mathematical Reviews (MathSciNet): MR728447

Zentralblatt MATH: 0529.60099

Digital Object Identifier: doi:10.1007/BF01206313

Project Euclid: euclid.cmp/1103940734

[2] Boivin, D. (1998). Ergodic theorems for surfaces with minimal random weights. Ann. Inst. H. Poincaré Probab. Statist. 34 567–599.

Mathematical Reviews (MathSciNet): MR1641662

Zentralblatt MATH: 0910.60078

Digital Object Identifier: doi:10.1016/S0246-0203(98)80001-0

[3] Cerf, R. (2006). The Wulff crystal in Ising and percolation models. In École d’été de probabilités de Saint-Flour, XXIV—2004. Lecture Notes in Mathematics 1878 268. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2241754

Zentralblatt MATH: 1103.82010

[4] Chayes, J. T. and Chayes, L. (1986). Bulk transport properties and exponent inequalities for random resistor and flow networks. Comm. Math. Phys. 105 133–152.

Mathematical Reviews (MathSciNet): MR847132

Zentralblatt MATH: 0617.60099

Digital Object Identifier: doi:10.1007/BF01212346

Project Euclid: euclid.cmp/1104115261

[5] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583–603.

Mathematical Reviews (MathSciNet): MR624685

Zentralblatt MATH: 0462.60012

Digital Object Identifier: doi:10.1214/aop/1176994364

Project Euclid: euclid.aop/1176994364

[6] Diestel, R. (2005). Graph Theory, 3rd ed. Graduate Texts in Mathematics 173. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2159259

[7] Estrada, F. J. and Jepson, A. D. (2005). Quantitative evaluation of a novel image segmentation algorithm. In Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05) IEEE Computer Society 2 1132–1139. Washington, DC, USA.

[8] Ford, L. R., Jr. and Fulkerson, D. R. (1956). Maximal flow through a network. Canad. J. Math. 8 399–404.

Mathematical Reviews (MathSciNet): MR79251

[9] Garet, O. and Marchand, R. (2007). Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35 833–866.

Mathematical Reviews (MathSciNet): MR2319709

Zentralblatt MATH: 1117.60090

Digital Object Identifier: doi:10.1214/009117906000000881

Project Euclid: euclid.aop/1178804316

[10] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439–457.

Mathematical Reviews (MathSciNet): MR1068308

Digital Object Identifier: doi:10.1098/rspa.1990.0100

JSTOR: links.jstor.org

[11] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1707339

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

Mathematical Reviews (MathSciNet): MR751574

[13] Kesten, H. (1987). Surfaces with minimal random weights and maximal flows: A higher-dimensional version of first-passage percolation. Illinois J. Math. 31 99–166.

Mathematical Reviews (MathSciNet): MR869483

Zentralblatt MATH: 0591.60096

Project Euclid: euclid.ijm/1255989405

[14] Lachand-Robert, T. and Oudet, É. (2005). Minimizing within convex bodies using a convex hull method. SIAM J. Optim. 16 368–379 (electronic).

Mathematical Reviews (MathSciNet): MR2197985

Zentralblatt MATH: 1104.65056

Digital Object Identifier: doi:10.1137/040608039

[15] Théret, M. (2006). On the small maximal flows in first passage percolation. Preprint. Available at http://arxiv.org/abs/math.PR/0607252.

Mathematical Reviews (MathSciNet): MR2464099

[16] Wu, X. (2006). Efficient algorithms for the optimal-ratio region detection problems in discrete geometry with applications. In Algorithms and Computation. Lecture Notes in Computer Science 4288 289–299. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR2296111

Digital Object Identifier: doi:10.1007/11940128_30

[17] Zhang, Y. (2000). Critical behavior for maximal flows on the cubic lattice. J. Statist. Phys. 98 799–811.

Mathematical Reviews (MathSciNet): MR1749233

Zentralblatt MATH: 0991.82019

Digital Object Identifier: doi:10.1023/A:1018631726709

previous :: next