Stable shredded spheres and causal random maps with large faces

TL;DR

This paper introduces stable shredded spheres, a new family of random metric spaces derived from stable Lévy processes, which serve as scaling limits for certain causal random planar maps with large faces.

Contribution

The paper constructs stable shredded spheres from stable Lévy excursions and establishes their role as scaling limits for models of causal random planar maps with large faces.

Findings

Stable shredded spheres are constructed from $ ext{Lévy}$ excursions.

They serve as scaling limits for causal random planar maps.

Their Hausdorff dimension equals $ ext{α}$.

Abstract

We introduce a new family of random compact metric spaces $\mathcal{S}_\alpha$ for $\alpha\in(1,2)$, which we call stable shredded spheres. They are constructed from excursions of $\alpha$-stable L\'evy processes on $[0,1]$ possessing no negative jumps. Informally, viewing the graph of the L\'evy excursion in the plane, each jump of the process is "cut open" and replaced by a circle and then all points on the graph at equal height which are not separated by a jump are identified. We show that the shredded spheres arise as scaling limits of models of causal random planar maps with large faces introduced by Di Francesco and Guitter. We also establish that their Hausdorff dimension is almost surely equal to $\alpha$. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive L\'evy processes as studied by Bertoin in the 90's.