The Mathematical Institute, University of Oxford, Eprints Archive

Heavy tails in last passage percolation

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 $\alpha<2$. 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 $\alpha$) of ``continuous last-passage percolation'' models in the unit square. In the extreme case $\alpha=0$ (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 $\alpha$-stable Levy processes, and indicate extensions of the results to higher dimensions.

Item Type:Article
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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