Unimodular hyperbolic planar graphs

We study random hyperbolic planar graphs by using their circle packing embedding to connect their geometry to that of the hyperbolic plane. This leads to several results: Identification of the Poisson and geometric boundaries, a connection between hyperbolicity and a form of non-amenability, and a new proof of the Benjamini-Schramm recurrence result. Based on works with subsets of Omer Angel, Martin Barlow, Ori Gurel-Gurevich, Tom Hutchcroft and Gourab Ray.