2006/04/08 by Ben Hambly, Hambly, Ben, Jean Martin +2
Computer Science · Mathematics · #60K35 #82B41 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35 #msc:82B41
paper · pdf · doi:10.48550/arxiv.math/0604189
43 pages, 6 figures
arxiv created 2006/04/08 · openalex publication_date 2006/04/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 R2 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.