Brownian geometry

TL;DR

This paper explores various continuous models of random geometry, focusing on the Brownian map and disk, which serve as universal scaling limits of large planar maps, highlighting the role of Brownian motion indexed by the Brownian tree.

Contribution

It reviews the construction and properties of Brownian geometry models, emphasizing the role of Brownian motion indexed by the Brownian tree in these models.

Findings

Brownian map as universal limit of large planar maps

Brownian disk as limit of planar maps with boundary

Role of Brownian motion indexed by the Brownian tree

Abstract

We present different continuous models of random geometry that have been introduced and studied in the recent years. In particular, we consider the Brownian map, which is the universal scaling limit of large planar maps in the Gromov-Hausdorff sense, and the Brownian disk, which appears as the scaling limit of planar maps with a boundary. We discuss the construction of these models, and we emphasize the role played by Brownian motion indexed by the Brownian tree.