Abstract
We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index α < 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 α) of “continuous last-passage percolation” models in the unit square. In the extreme case α = 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 α-stable Lévy processes, and indicate extensions of the results to higher dimensions.
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Hambly, B., Martin, J.B. Heavy tails in last-passage percolation. Probab. Theory Relat. Fields 137, 227–275 (2007). https://doi.org/10.1007/s00440-006-0019-0
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DOI: https://doi.org/10.1007/s00440-006-0019-0
Keywords
- Last-passage percolation
- Heavy tails
- Airy process
- Regular variation
- Stable process
- Multifractal spectrum
Mathematics Subject Classifications (2000)
- Primary 60K35
- Secondary 82B41