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Lower large deviations for the maximal flow through a domain of \({\mathbb{R}^d}\) in first passage percolation
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  • Published: 28 April 2010

Lower large deviations for the maximal flow through a domain of \({\mathbb{R}^d}\) in first passage percolation

  • Raphaël Cerf1 &
  • Marie Théret2 

Probability Theory and Related Fields volume 150, pages 635–661 (2011)Cite this article

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  • 10 Citations

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Abstract

We consider the standard first passage percolation model in the rescaled graph \({\mathbb{Z}^d/n}\) for d ≥ 2, and a domain Ω of boundary Γ in \({\mathbb{R}^d}\) . Let Γ1 and Γ2 be two disjoint open subsets of Γ, representing the parts of Γ through which some water can enter and escape from Ω. We investigate the asymptotic behaviour of the flow \({\phi_n}\) through a discrete version Ω n of Ω 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, the lower large deviations of \({\phi_n/ n^{d-1}}\) below a certain constant are of surface order.

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Authors and Affiliations

  1. Département de Mathématiques, Université Paris Sud, Bâtiment 425, 91405, Orsay Cedex, France

    Raphaël Cerf

  2. Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d’Ulm, 75230, Paris Cedex 05, France

    Marie Théret

Authors
  1. Raphaël Cerf
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  2. Marie Théret
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Correspondence to Marie Théret.

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Cerf, R., Théret, M. Lower large deviations for the maximal flow through a domain of \({\mathbb{R}^d}\) in first passage percolation. Probab. Theory Relat. Fields 150, 635–661 (2011). https://doi.org/10.1007/s00440-010-0287-6

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  • Received: 31 July 2009

  • Revised: 03 March 2010

  • Published: 28 April 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0287-6

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Keywords

  • First passage percolation
  • Maximal flow
  • Minimal cut
  • Large deviations

Mathematics Subject Classification (2000)

  • 60K35
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