Growth-fragmentations, Brownian cone excursions and SLE(6) explorations of a quantum disc

TL;DR

This paper explores the connection between growth-fragmentation processes, Brownian cone excursions, and SLE(6) explorations in quantum discs, using Brownian motion techniques to reveal new properties and implications for quantum boundary length dynamics.

Contribution

It introduces a growth-fragmentation process linked to Brownian cone excursions and demonstrates its relation to SLE(6) explorations in quantum discs using Brownian motion methods.

Findings

Identification of a growth-fragmentation process in Brownian cone excursions.

Connection between growth-fragmentation and quantum boundary length in SLE(6) explorations.

New properties of Brownian cone excursions and their links to stable Lévy processes.

Abstract

The aim of this article is to present a growth-fragmentation process naturally embedded in a Brownian excursion from boundary to apex in a cone of angle $2\pi/3$. This growth-fragmentation process corresponds, via the so-called mating-of-trees encoding arXiv:1409.7055, to the quantum boundary length process associated with a branching $\mathrm{SLE}_6$ exploration of a $\gamma= \sqrt{8/3}$ quantum disc. However, our proof uses only Brownian motion techniques, and along the way we discover various properties of Brownian cone excursions and their connections with stable L\'evy processes. Assuming the mating of trees encoding, our results imply several fundamental properties of the ${\gamma}= \sqrt{8/3}$-quantum disc $\mathrm{SLE}_6$-exploration.