Asymptotics for rooted planar maps and scaling limits of two-type spatial trees

TL;DR

This paper establishes asymptotic properties of large bipartite planar maps, including radius and profile, by linking them to two-type spatial trees and analyzing their scaling limits.

Contribution

It introduces new asymptotic results for bipartite planar maps using a bijection with two-type trees and a limit theorem for spatial trees.

Findings

Asymptotic behavior of radius and profile of large bipartite planar maps.

High probability of existence of separating vertices in large bipartite maps.

Connection between planar maps and two-type spatial trees for asymptotic analysis.

Abstract

We prove some asymptotic results for the radius and the profile of large random bipartite planar maps. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted bipartite planar maps and certain two-type trees with positive labels, we derive our results from a conditional limit theorem for two-type spatial trees. Finally we apply our estimates to separating vertices of bipartite planar maps: with probability close to one when $n$ goes to infinity, a random $2\ka$-angulation with $n$ faces has a separating vertex whose removal disconnects the map into two components each with size greater that $n^{1/2-\vep}$.