Growing random 3-connected maps, or comment s'enfuir de l'hexagone

TL;DR

This paper introduces a local growth procedure for 3-connected maps using bijections with binary trees and quadrangulations, demonstrating convergence to a limit distribution related to large random 3-connected graphs.

Contribution

It develops a novel local growth process for maps based on bijections, and establishes the limit distribution for large random 3-connected maps.

Findings

Probability of the nth map being 3-connected tends to 2^8/3^6.

Sequence of maps converges to an almost sure limit G.

G is the distributional local limit of large random 3-connected graphs.

Abstract

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. As n tends to infinity, the probability that the n'th map in the sequence is 3-connected tends to 2^8/3^6. The sequence of maps has an almost sure limit G, and we show that G is the distributional local limit of large, uniformly random 3-connected graphs.