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arXiv · 1204.5921

The Brownian plane

Nicolas Curien·Jean-François Le Gall
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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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Nicolas Curien, Jean-François Le Gall. 2012-04-26. The Brownian plane. https://arxiv.org/abs/1204.5921

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