On the Riemann surface type of random planar maps

  • James T. Gill

    Saint Louis University, St. Louis, USA
  • Steffen Rohde

    University of Washington, Seattle, USA


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.

Cite this article

James T. Gill, Steffen Rohde, On the Riemann surface type of random planar maps. Rev. Mat. Iberoam. 29 (2013), no. 3, pp. 1071–1090

DOI 10.4171/RMI/749