The snake in the Brownian sphere

TL;DR

This paper establishes a measurable inverse mapping from the Brownian sphere to the Brownian snake, enhancing understanding of the continuous analogue of the CVS bijection in random planar maps.

Contribution

It constructs the inverse of the continuous CVS bijection, linking the Brownian sphere and the Brownian snake in a measurable way.

Findings

Constructed the inverse of the continuous CVS bijection.

Demonstrated the measurable dependence of the Brownian snake on the Brownian sphere.

Addressed orientation issues in the Brownian sphere.

Abstract

The Brownian sphere is a random metric space, homeomorphic to the two-dimensional sphere, which arises as the universal scaling limit of many types of random planar maps. The direct construction of the Brownian sphere is via a continuous analogue of the Cori--Vauquelin--Schaeffer (CVS) bijection. The CVS bijection maps labeled trees to planar maps, and the continuous version maps Aldous' continuum random tree with Brownian labels (the Brownian snake) to the Brownian sphere. In this work, we describe the inverse of the continuous CVS bijection, by constructing the Brownian snake as a measurable function of the Brownian sphere. Special care is needed to work with the orientation of the Brownian sphere.