Random cubic planar maps

TL;DR

This paper studies uniform random cubic planar maps, deriving their limiting distributions and enumeration formulas, revealing new asymptotic behaviors for parameters like the largest block and 3-connected components.

Contribution

It introduces a unified enumeration approach for cubic planar maps and establishes new limiting distribution laws for key structural parameters.

Findings

Distribution of root face degree has an exponential tail.

Largest block size follows a map-Airy distribution with expectation ~n/√3.

Largest 3-connected component has expected size ~n/4.

Abstract

We analyse uniform random cubic rooted planar maps and obtain limiting distributions for several parameters of interest. From the enumerative point of view, we present a unified approach for the enumeration of several classes of cubic planar maps, which allow us to recover known results in a more general and transparent way. This approach allows us to obtain new enumerative results. Concerning random maps, we first obtain the distribution of the degree of the root face, which has an exponential tail as for other classes of random maps. Our main result is a limiting map-Airy distribution law for the size of the largest block $L$, whose expectation is asymptotically $n/\sqrt{3}$ in a random cubic map with $n+2$ faces. We prove analogous results for the size of the largest cubic block, obtained from $L$ by erasing all vertices of degree two, and for the size of the largest 3-connected component, whose expected values are respectively $n/2$ and $n/4$. To obtain these results we need to analyse a new type of composition scheme which has not been treated by Banderier et al. [Random Structures Algorithms 2001].