Scaling limits of random planar maps with a unique large face

TL;DR

This paper investigates the asymptotic behavior of large random bipartite planar maps with a dominant large face, showing convergence to the Brownian tree and describing the distance profile via Brownian excursions.

Contribution

It establishes the emergence of a unique large face in large maps and proves their convergence to the Brownian tree in the Gromov-Hausdorff sense.

Findings

Presence of a unique large face proportional to the map size

Distance profiles converge to a Brownian excursion

Rescaled maps converge to Aldous' Brownian tree

Abstract

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges $n$ of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.