Stable shredded spheres and causal random maps with large faces

Abstract

We introduce a new familiy of random compact metric spaces ${\mathcal{S}}_{\mathit{\alpha }}$ for $\mathit{\alpha }\in (1,2)$, which we call stable shredded spheres. They are constructed from excursions of α-stable Lévy processes on $[0,1]$ possessing no negative jumps. Informally, viewing the graph of the Lévy 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 α. Point identification in the shredded spheres is intimately connected to the presence of decrease points in stable spectrally positive Lévy processes, as studied by Bertoin in the 1990s.