To what extent does the random walk on the circle packing of a plane triangulation behave like Brownian motion? We show that with a certain natural assignment of weights to the edges (due to Dubejko 1995) the exit measure of the walk from a given domain converges to the exit measure of Brownian motion as the circles get smaller. This improves previous results of Skopenkov 2013 and Werness 2015. An important point is that no assumptions on the vertex degrees of the map is made, making the result applicable to random planar maps models.
Joint work with O. Gurel-Gurevich and D. Jerison