Asymptotic Distribution of Parameters in Random Maps

TL;DR

This paper investigates the limiting distributions of six parameters in large random rooted maps, revealing a variety of distributions including geometric, Poisson, Beta, normal, uniform, and a novel recursive distribution.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of multiple parameters in random maps, identifying their distinct limiting distributions.

Findings

Vertices follow a Poisson distribution.

Root vertex degree converges to a normal distribution.

Some parameters exhibit a Beta or uniform distribution in the limit.

Abstract

We consider random rooted maps without regard to their genus, with fixed large number of edges, and address the problem of limiting distributions for six different parameters: vertices, leaves, loops, root edges, root isthmus, and root vertex degree. Each of these leads to a different limiting distribution, varying from (discrete) geometric and Poisson distributions to different continuous ones: Beta, normal, uniform, and an unusual distribution whose moments are characterised by a recursive triangular array.