Spatial Markov property in Brownian disks

TL;DR

This paper introduces a new representation of the Brownian disk using labeled trees, establishing a spatial Markov property that describes the independence of the disk's complement of a hull, with implications for peeling processes.

Contribution

It provides a novel forest-based representation of the Brownian disk and proves a spatial Markov property, advancing understanding of its boundary and hull structures.

Findings

Representation of Brownian disk via labeled trees

Spatial Markov property for hulls in Brownian disks

Independence of hull complements conditioned on perimeter

Abstract

We derive a new representation of the Brownian disk in terms of a forest of labeled trees, where labels correspond to distances from a subset of the boundary. We then use this representation to obtain a spatial Markov property showing that the complement of a hull centered at a boundary point of a Brownian disk is again a Brownian disk, with a random perimeter, and is independent of the hull conditionally on its perimeter. Our proofs rely in part on a study of the peeling process for triangulations with a boundary, which is of independent interest. The results of the present work will be applied to a continuous version of the peeling process for the Brownian half-plane in a companion paper.