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Transactions of the American Mathematical Society

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Law of large numbers for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation


Authors: Raphaël Cerf and Marie Théret
Journal: Trans. Amer. Math. Soc. 363 (2011), 3665-3702
MSC (2010): Primary 60K35, 49Q20
DOI: https://doi.org/10.1090/S0002-9947-2011-05341-9
Published electronically: February 25, 2011
MathSciNet review: 2775823
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Abstract: We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq 2$, and a domain $\Omega$ of boundary $\Gamma$ in $\mathbb {R}^d$. Let $\Gamma ^1$ and $\Gamma ^2$ be two disjoint open subsets of $\Gamma$, representing the parts of $\Gamma$ through which some water can enter and escape from $\Omega$. We investigate the asymptotic behaviour of the flow $\phi _n$ through a discrete version $\Omega _n$ of $\Omega$ between the corresponding discrete sets $\Gamma ^1_n$ and $\Gamma ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, $\phi _n$ converges almost surely towards a constant $\phi _{\Omega }$, which is the solution of a continuous non-random min-cut problem. Moreover, we give a necessary and sufficient condition on the law of the capacity of the edges to ensure that $\phi _{\Omega } >0$.


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Additional Information

Raphaël Cerf
Affiliation: Mathématique, Université Paris Sud, bâtiment 425, 91405 Orsay Cedex, France
MR Author ID: 349311
Email: rcerf@math.u-psud.fr

Marie Théret
Affiliation: DMA, École Normale Supérieure, 45 rue d’Ulm, 75230 Paris Cedex 05, France
Address at time of publication: LPMA, Université Paris Diderot Site Chevaleret, Case 7012, 75205 Paris Cedex 12 France
Email: marie.theret@ens.fr, marie.theret@univ-paris-diderot.fr

Keywords: First passage percolation, maximal flow, minimal cut, law of large numbers, polyhedral approximation.
Received by editor(s): November 5, 2009
Published electronically: February 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.