The continuous limit of large random planar maps

Jean-François Le Gall

doi:10.46298/dmtcs.3554

Abstract

We discuss scaling limits of random planar maps chosen uniformly over the set of all $2p$-angulations with $n$ faces. This leads to a limiting space called the Brownian map, which is viewed as a random compact metric space. Although we are not able to prove the uniqueness of the distribution of the Brownian map, many of its properties can be investigated in detail. In particular, we obtain a complete description of the geodesics starting from the distinguished point called the root. We give applications to various properties of large random planar maps.

Highlights

The main purpose of the present article is to survey recent developments about scaling limits of large planar maps chosen uniformly at random in a suitable class
The Brownian map is described as a quotient space of the continuous random tree called the CRT, for an equivalence relation that is defined in terms of Brownian labels assigned to the vertices of the CRT
If the previous theorem provides a candidate for the scaling limit of p-trees, we still need to introduce the analogue of labels in the continuous setting

Summary

Introduction

The main purpose of the present article is to survey recent developments about scaling limits of large planar maps chosen uniformly at random in a suitable class. As a consequence of the previous discussion, it makes sense to study the convergence in distribution of the random metric spaces (1) as random variables with values in the Polish space (K, dGH ) This problem was stated in this form for triangulations by Schramm (Sc), but the general idea of finding a continuous limit for large random planar maps had appeared earlier, in particular in the pioneering paper of Chassaing and Schaeffer (CS). The Brownian map is described as a quotient space of the continuous random tree called the CRT, for an equivalence relation that is defined in terms of Brownian labels assigned to the vertices of the CRT (see Sections 3 and 4 below for a detailed discussion) This space and its topological structure are completely determined, the distance on the Brownian map has not been characterized, and this explains the need for a subsequence in our limit theorem for the sequence (V (Mn), n−1/4dgr).

Bijections betwen maps and trees

Scaling limits of trees

Convergence towards the Brownian map

Two theorems about the Brownian map

Geodesics in the Brownian map