The Brownian Plane

Nicolas Curien and Jean-François Le Gall ()
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Nicolas Curien: Ecole normale supérieure
Jean-François Le Gall: Université Paris-Sud

Journal of Theoretical Probability, 2014, vol. 27, issue 4, 1249-1291

Abstract: Abstract We introduce and study the random non-compact metric space called the Brownian plane, which is obtained as the scaling limit of the uniform infinite planar quadrangulation. Alternatively, the Brownian plane is identified as the Gromov–Hausdorff tangent cone in distribution of the Brownian map at its root vertex, and it also arises as the scaling limit of uniformly distributed (finite) planar quadrangulations with $$n$$ faces when the scaling factor tends to $$0$$ less fast than $$n^{-1/4}$$ . We discuss various properties of the Brownian plane. In particular, we prove that the Brownian plane is homeomorphic to the plane, and we get detailed information about geodesic rays to infinity.

Keywords: Random planar map; Brownian map; Brownian plane; Uniform infinite planar quadrangulation; Gromov–Hausdorff convergence; Scaling limit; Primary: 05C80; 60D05; Secondary: 05C12; 60F17 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-013-0485-0

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