Revista Matemática Iberoamericana
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Published online: 2013-08-04
On the Riemann surface type of random planar mapsJames T. Gill and Steffen Rohde (1) Saint Louis University, St. Louis, USA
(2) 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.
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Gill James, Rohde Steffen: On the Riemann surface type of random planar maps. Rev. Mat. Iberoamericana 29 (2013), 1071-1090. doi: 10.4171/RMI/749