The Brownian plane

TL;DR

The paper introduces the Brownian plane, a new random non-compact metric space as a scaling limit of infinite planar quadrangulations, and explores its properties including topology and geodesics.

Contribution

It defines and analyzes the Brownian plane, establishing its relation to the Brownian map and finite quadrangulations, and details its topological and geometric properties.

Findings

Brownian plane is homeomorphic to the plane

It is the Gromov-Hausdorff tangent cone of the Brownian map

Provides detailed descriptions of geodesic rays to infinity

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.