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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Law of large numbers for the maximal flow through a domain of $\mathbb {R}^d$ in first passage percolation
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by Raphaël Cerf and Marie Théret PDF
Trans. Amer. Math. Soc. 363 (2011), 3665-3702 Request permission


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:
  • 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:,
  • Received by editor(s): November 5, 2009
  • Published electronically: February 25, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 3665-3702
  • MSC (2010): Primary 60K35, 49Q20
  • DOI:
  • MathSciNet review: 2775823