## Sigurður Örn Stefánsson (30/10/17)

Math Colloquium

### Speaker: Sigurður Örn Stefánsson, University of Iceland

### Title: The phase structure of random outerplanar maps

Location: VRII-158

Time: Monday 30 October at 15:00

### Abstract:

An outerplanar map is a drawing of a planar graph in the sphere which has the property that there is a face in the map such that all the vertices lie on the boundary of that face. We study the phase diagram of random outerplanar maps sampled according to non-negative weights that are assigned to each face of a map. We prove that for certain choices of weights the map looks like a rescaled version of its boundary when its number of vertices tends to infinity. The outerplanar maps are then shown to converge in the Gromov-Hausdorff sense towards the α-stable looptree introduced by Curien and Kortchemski (2014), with the parameter α depending on the specific weight-sequence. This allows us to describe the transition of the asymptotic geometric shape from a deterministic circle to the Brownian tree.

Based on arXiv:1710.04460 with Benedikt Stufler.