Cut Vertices in Random Planar Maps

TL;DR

This paper investigates the asymptotic behavior of cut-vertices in random planar maps, establishing their proportional growth and a central limit theorem for certain subclasses, revealing complex combinatorial structures.

Contribution

It provides the first asymptotic analysis of cut-vertices in random planar maps, including explicit constants and CLT results for subcritical classes.

Findings

Number of cut-vertices grows proportionally with edges

Convergence in probability to a constant c>0

Central limit theorem for outerplanar maps

Abstract

The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n \to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.