Unimodular hyperbolic triangulations
For deterministic bounded degree triangulations, circle packing has proven a powerful tool for studying random walk via geometric arguments. In this talk, I will discuss extensions and analogues for random triangulations without the assumption of bounded degree. In particular, I will show that the circle packing type (hyperbolic or Euclidean) is determined by the expected degree at the root and that, in the hyperbolic case, the geometric boundary given by the circle packing coincides with the Poisson boundary of the random walk. No specialised knowledge will be assumed and I will review the main examples.
Joint work with Omer Angel, Asaf Nachmias and Gourab Ray.