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 $α\in(1,2)$. 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 $α$. In particular, the profile of distances in the map, rescaled by the factor $n^{-1/2α}$, 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 $2α$.