On the Riemann surface type of random planar maps

Abstract

We show that the (random) Riemann surfaces of the Angel-Schramm uniform infinite planar triangulation and of Sheffield's infinite necklace construction are both parabolic. In other words, Brownian motion on these surfaces is recurrent. We obtain this result as a corollary to a more general theorem on subsequential distributional limits of random unbiased disc triangulations, following work of Benjamini and Schramm. © European Mathematical Society.