The Brownian Plane

By Nicolas Curien, Jean-François Le Gall. 2012

97 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/04e31ee1b9e09feab836bdb8259b23dd3a92c320

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Liouville Quantum Gravity on the Riemann Sphere (2014, background): “This metric space is supposed to correspond to the conformal embedding in the plane of the so-called Brownian plane, recently constructed in [13].”
Planar stochastic hyperbolic triangulations (2016, methodology): “Contrary to the case of the UIPT which is believed to converge towards the Brownian plane [20] in the scaling limit for the Gromov–Hausdorff topology, it is pretty clear that in the hyperbolic regime Tκ do not admit any scaling limits in the Gromov–Hausdorff sense3.”
Random geometry on the sphere (2014, methodology): “vich and Nachmias [26]. See also Benjamini and Curien [10] for a proof of subdiusivity of simple random walk on the UIPQ. 5.2. Convergence to the Brownian plane. The following results are taken from [21]. To simplify notation, we write Q 1= M (4) for the UIPQ, which was introduced in the previous su”
Compact Brownian surfaces I: Brownian disks (2015, background): “e two facts indeed qualify the Brownian map as being a canonical random geometry on thesphere. Notethatanon-compactvariantoftheBrownianmap,calledthe Brownian 1 INTRODUCTION 3 plane,hasbeenintroducedin[CLG12]andshowntobethescalinglimitofsomenatural modelsofrandomquadrangulations. However,forothertopo”
The hull process of the Brownian plane (2014, methodology): “The probabilistic structure of the real tree T∞ is more complicated than in [9], but the labels are now nonnegative and correspond to distances from the root in P∞ (whereas in [9] labels corresponded in some sense to “distances from infinity”).”
Scaling limits for the peeling process on random maps (2014, background): “(62) The transitions in the peeling process of the UIPQ are more complicated than previously because of additional cases.”