The CRT is the scaling limit of random dissections

TL;DR

This paper demonstrates that large random polygon dissections, when scaled appropriately, converge to a Brownian tree, revealing a universal scaling limit influenced by Boltzmann weights.

Contribution

It establishes the Gromov--Hausdorff convergence of scaled random dissections to a Brownian tree and computes the scaling constant based on Boltzmann weights.

Findings

Random dissections converge to a Brownian tree in the scaling limit.

The scaling constant depends on Boltzmann weights in a specific way.

The convergence holds for various types of dissections, including uniform cases.

Abstract

We study the graph structure of large random dissections of polygons sampled according to Boltzmann weights, which encompasses the case of uniform dissections or uniform $p$-angulations. As their number of vertices $n$ goes to infinity, we show that these random graphs, rescaled by $n^{-1/2}$, converge in the Gromov--Hausdorff sense towards a multiple of Aldous' Brownian tree when the weights decrease sufficiently fast. The scaling constant depends on the Boltzmann weights in a rather amusing and intriguing way, and is computed by making use of a Markov chain which compares the length of geodesics in dissections with the length of geodesics in their dual trees.