Heavy tails in last-passage percolation

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 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.