Drilling holes in the Brownian disk: The Brownian annulus

TL;DR

This paper introduces a new construction of the Brownian annulus via hull removal in the free Brownian disk, demonstrating it as a scaling limit of Boltzmann triangulations with two boundaries and analyzing related structures.

Contribution

It provides a novel construction of the Brownian annulus and establishes its relation as a scaling limit of Boltzmann triangulations with two boundaries.

Findings

Brownian annulus constructed by hull removal in the Brownian disk

Brownian annulus is the scaling limit of Boltzmann triangulations with two boundaries

Analysis of peeling by layers algorithm for Boltzmann triangulations

Abstract

We give a new construction of the Brownian annulus based on removing a hull centered at the distinguished point in the free Brownian disk. We use this construction to prove that the Brownian annulus is the scaling limit of Boltzmann triangulations with two boundaries. We also prove that the space obtained by removing hulls centered at the two distinguished points of the Brownian sphere is a Brownian annulus. Our proofs rely on a detailed analysis of the peeling by layers algorithm for Boltzmann triangulations with a boundary.