The Brownian Cactus I. Scaling limits of discrete cactuses

TL;DR

This paper proves that the discrete cactuses of large random planar maps converge to a continuous limit called the Brownian cactus, which is related to the Brownian map, in the Gromov-Hausdorff sense.

Contribution

It establishes the convergence of discrete cactuses of random planar maps to the Brownian cactus, linking discrete structures to continuous limits.

Findings

Discrete cactuses converge to the Brownian cactus in distribution.

The Brownian cactus is the continuous cactus of the Brownian map.

Results hold under general assumptions for Boltzmann-weighted maps.

Abstract

The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\R$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.