Constructing the Brownian sphere from a continuum random unicycle

TL;DR

This paper presents a new explicit construction of the Brownian sphere with two marked points at a fixed distance, utilizing the Miermont bijection, and explores related conditionings and representations of related random geometric objects.

Contribution

It introduces a novel construction of the Brownian sphere biased by distance and connects it to Voronoi cells and the bigeodesic Brownian plane.

Findings

Explicit construction of the Brownian sphere with two distinguished points.

New representation of the bigeodesic Brownian plane.

Connections between Brownian sphere conditionings and Voronoi cells.

Abstract

We give an explicit construction of the Brownian sphere biased by the distance between two distinguished points, which is based on the Miermont bijection for quadrangulations. We then describe various conditionings of this object, which are related to Vorono\"i cells in the Brownian sphere. In particular, we give a new construction of the Brownian sphere with two distinguished points at a fixed distance. We also use this construction to derive a new representation of the bigeodesic Brownian plane.