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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
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

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

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.