Radius and profile of random planar maps with faces of arbitrary degrees

TL;DR

This paper investigates the asymptotic behavior of the radius and profile of large random rooted planar maps with arbitrary face degrees, utilizing bijections to analyze associated labeled trees.

Contribution

It introduces new asymptotic results for the radius and profile of such maps, based on a novel application of bijections and limit theorems for labeled trees.

Findings

Asymptotic distribution of the radius of large maps

Profile characteristics of large random planar maps

Application of four-type spatial Galton-Watson trees in analysis

Abstract

We prove some asymptotic results for the radius and the profile of large random rooted planar maps with faces of arbitrary degrees. Using a bijection due to Bouttier, Di Francesco and Guitter between rooted planar maps and certain four-type trees with positive labels, we derive our results from a conditional limit theorem for four-type spatial Galton-Watson trees.