Hambly, B. M. and Martin, J. B. (2005) *Heavy tails in last passage percolation.* probability theory and related fields . (In Press)

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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 . 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 ) of ``continuous last-passage percolation'' models in the unit square. In the extreme case (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 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 -stable Levy processes, and indicate extensions of the results to higher dimensions.

Item Type: | Article |
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Subjects: | O - Z > Probability theory and stochastic processes |

Research Groups: | Stochastic Analysis Group Oxford Centre for Industrial and Applied Mathematics |

ID Code: | 289 |

Deposited By: | Ben Hambly |

Deposited On: | 02 Oct 2006 |

Last Modified: | 20 Jul 2009 14:20 |

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