Local convergence of large critical multi-type Galton-Watson trees and applications to random maps

TL;DR

This paper proves that large critical multi-type Galton-Watson trees converge locally to an infinite tree and applies this to show that large critical random maps also converge to an infinite map, advancing understanding of their asymptotic structure.

Contribution

It establishes local convergence results for large multi-type Galton-Watson trees and applies these to analyze the local limits of large random planar maps.

Findings

Large critical multi-type Galton-Watson trees converge locally to an infinite tree.

Large critical Boltzmann-distributed random maps converge to an infinite map.

Results extend the understanding of asymptotic structures in random combinatorial objects.

Abstract

We show that large critical multi-type Galton-Watson trees, when conditioned to be large, converge locally in distribution to an infinite tree which is analoguous to Kesten's infinite monotype Galton-Watson tree. This is proven when we condition on the number of vertices of one fixed types, and with an extra technical assumption if we count at least two types. We then apply these results to study local limits of random planar maps, showing that large critical Boltzmann-distributed random maps converge in distribution to an infinite map.