Scaling limits of random planar maps with large faces

Abstract

We discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index $\alpha \in (1,2)$. When the number $n$ of vertices of the map tends to infinity, the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process, which can be constructed from a centered stable process with no negative jumps and index $\alpha$. In particular, the profile of distances in the map, rescaled by the factor $n^{-1/2\alpha }$, converges to a random measure defined in terms of the distance process. With the same rescaling of distances, the vertex set viewed as a metric space converges in distribution as $n\to \infty$, at least along suitable subsequences, toward a limiting random compact metric space whose Hausdorff dimension is equal to $2\alpha$.

arXiv commentPublished in at http://dx.doi.org/10.1214/10-AOP549 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)