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On the Riemann surface type of Random Planar Maps · Around

2011

On the Riemann surface type of Random Planar Maps

Gill, James T., Rohde, Steffen

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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.

Built on

Z. X. He, O. Schramm, Hyperbolic and Parabolic Packings
1995
Earlier work this paper cites.
O. Schramm, Transboundary Extremal Length
1995
Earlier work this paper cites.
G. McCaughan, A recurrence/transience result for circle packings
1998
Earlier work this paper cites.
G. Schaeffer, Conjugaison d’arbres et cartes combinatoires alatoires
1998
Earlier work this paper cites.

Similar

I. Benjamini, O. Schramm, Recurrence of Distributional Limits of Finite Planar Graphs
2001
Cited alongside, same era.
O. Angel, O. Schramm, Uniform infinite planar triangulations, Comm. Math. Phys. 241
2003
Cited alongside, same era.
J. Bouttier, P. Di Francesco, E. Guitter, Planar maps as labeled mobiles
2004
Cited alongside, same era.
K. Stephenson, Introduction to Circle Packing: The Theory of Discrete Analytic Functions
Original
2005
Cited alongside, same era.
K. Astala, P. Jones, A. Kupiainen, E. Saksman, Random conformal weldings
Cited in the paper.
I. Benjamini, Random Planar Metrics
Cited in the paper.
I. Benjamini, N. Curien, On limits of Graphs Sphere Packed in Euclidean Space and Applications
Original
Cited in the paper.
B. Duplantier, S. Sheffield, Liouville Quantum Gravity and KPZ
Cited in the paper.
S. Rohde, Oded Schramm: From Circle Packings to SLE
Cited in the paper.
S. Sheffield, Conformal weldings of random surfaces: SLE and the quantum gravity zipper
Original
Cited in the paper.

Then

J.-F. Le Gall, The topological structure of scaling limits of planar maps
2007
Later among the works it cites.
J.-F. Le Gall, F. Paulin, Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere
2008
Later among the works it cites.
G. Miermont, On the sphericity of scaling limits of random planar quadrangulations
2008
Later among the works it cites.

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