Scaling limits of random planar maps with large faces

TL;DR

This paper investigates the asymptotic geometric properties of large random planar maps with faces having degrees in the domain of attraction of a stable law, revealing a limiting fractal structure characterized by a stable process.

Contribution

It introduces a new scaling limit for random planar maps with large faces, connecting their geometry to stable processes and describing the limiting metric space.

Findings

Distances rescaled by n^{-1/2α} converge to a continuous distance process

The profile of distances converges to a random measure based on the distance process

The vertex set converges to a fractal-like limiting space with Hausdorff dimension 2α

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 $2\alpha$.