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The Brownian Plane

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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.

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Acknowledgments

We are indebted to Grégory Miermont for a number of very useful discussions.

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Authors and Affiliations

  1. Ecole normale supérieure, Paris, France

    Nicolas Curien

  2. Université Paris-Sud, Orsay, France

    Jean-François Le Gall

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  1. Nicolas Curien
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  2. Jean-François Le Gall
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Correspondence to Jean-François Le Gall.

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Curien, N., Le Gall, JF. The Brownian Plane. J Theor Probab 27, 1249–1291 (2014). https://doi.org/10.1007/s10959-013-0485-0

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  • DOI: https://doi.org/10.1007/s10959-013-0485-0

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