Speaker:Joonas Turunen
Title:Scaling limits of random planar maps with large faces and a causal structure

Abstract:
Planar maps are proper embeddings of finite connected planar (multi)graphs to the 2-sphere. In addition to being interesting as objects in combinatorial probability, random planar maps have been widely studied due to their role as a discretization of the random surfaces in the Liouville quantum gravity. A landmark result in this field is the theorem proven by Le Gall and Miermont, which says that numerous collections of random planar maps with bounded face degrees converge in distribution to a random compact metric space, the Brownian Map, in the sense of Gromov-Hausdorff when the graph distance is suitably rescaled. Motivated by random planar maps decorated by statistical mechanics models, Le Gall and Miermont also introduced a planar map model where a typical face degree is sampled from a heavy-tailed distribution, and showed that such maps suitably rescaled converge along subsequences to random compact metric spaces called stable maps.

Recently, Björnberg, Curien and Stefánsson studied a similar model where the maps possess a causal structure in addition to the presence of large faces, and showed that such maps converge to a collection of unique scaling limits called the stable shredded spheres. Such causal maps are interesting due to their connections to the Lorentzian quantum gravity. In this talk, I will discuss the aforementioned objects, and introduce a novel collection of random planar maps in the presence of both large faces and a causal structure, whose scaling limits interpolate between the stable shredded spheres and a continuum random tree model. In some sense, the geometry of such limiting random compact metric spaces is a mixture of the stable shredded spheres and random continuum trees. Based on a joint ongoing work with Jakob Björnberg and Sigurður Örn Stefánsson