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Heavy tails in last-passage percolation
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  • Published: 20 September 2006

Heavy tails in last-passage percolation

  • Ben Hambly1 &
  • James B. Martin2 

Probability Theory and Related Fields volume 137, pages 227–275 (2007)Cite this article

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  • 17 Citations

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Abstract

We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index α < 2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by α) of “continuous last-passage percolation” models in the unit square. In the extreme case α = 0 (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to \(\mathbb{R}^2\) we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on α-stable Lévy processes, and indicate extensions of the results to higher dimensions.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford, OX1 3LB, UK

    Ben Hambly

  2. Department of Statistics, University of Oxford, 1 South Parks Road, Oxford, OX1 3TG, UK

    James B. Martin

Authors
  1. Ben Hambly
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  2. James B. Martin
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Corresponding author

Correspondence to James B. Martin.

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Cite this article

Hambly, B., Martin, J.B. Heavy tails in last-passage percolation. Probab. Theory Relat. Fields 137, 227–275 (2007). https://doi.org/10.1007/s00440-006-0019-0

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  • Received: 19 July 2005

  • Revised: 08 April 2006

  • Published: 20 September 2006

  • Issue Date: January 2007

  • DOI: https://doi.org/10.1007/s00440-006-0019-0

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Keywords

  • Last-passage percolation
  • Heavy tails
  • Airy process
  • Regular variation
  • Stable process
  • Multifractal spectrum

Mathematics Subject Classifications (2000)

  • Primary 60K35
  • Secondary 82B41
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