The bigeodesic Brownian plane

TL;DR

The paper introduces the bigeodesic Brownian plane, a new random geometric space, and demonstrates its role as a tangent in distribution to the Brownian sphere and as a local limit of the Brownian plane rerooted along its geodesic.

Contribution

It defines and analyzes the properties of the bigeodesic Brownian plane, establishing its significance as a tangent space and local limit in the context of random geometric structures.

Findings

Bigeodesic Brownian plane is the tangent in distribution of the Brownian sphere at a point on its geodesic.

It is the local limit of the Brownian plane rerooted along its infinite geodesic.

The space exhibits invariance under natural transformations and has well-characterized topological and geodesic properties.

Abstract

We introduce and study a random non-compact space called the bigeodesic Brownian plane, and prove that it is the tangent plane in distribution of the Brownian sphere at a point of its simple geodesic from the root (for the local Gromov-Hausdorff-Prokhorov-Uniform topology). We also show that it is the local limit of the Brownian plane rerooted further and further on its unique infinite geodesic. Furthermore, we discuss various properties of this space, such as its topology, the behavior of its geodesic rays, and its invariance in distribution under several natural transformations.