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: | 29 May 2015 18:19 |
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