Understand
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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Kesten, H
1986
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Bingham, N.H., Goldie, C.M., Teugels, J.L
1989
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Aldous, D
1991
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Revuz, D., Yor, M
1991
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1993
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Lyons, R., Peres, Y., Pemantle, R
1995
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