  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.
