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 lastpassage 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 lastpassage 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 selfsimilar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous lastpassage percolation model to we obtain a heavytailed 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 

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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