Local limits of large Galton-Watson trees rerooted at a random vertex

TL;DR

This paper studies the local structure of large random trees generated by Galton-Watson processes, revealing different limiting behaviors depending on the regime, including classical, condensation, and novel limit trees.

Contribution

It characterizes the local limits of large Galton-Watson trees at a random vertex across various regimes, including new limit trees in the condensation regime.

Findings

In the critical case, the limit is Aldous's invariant sin-tree.

In the condensation regime, the local structure is described up to the first large-degree ancestor.

A new limit tree is identified in the complete condensation subregime, including power-law offspring distributions.

Abstract

We discuss various forms of convergence of the vicinity of a uniformly at random selected vertex in random simply generated trees, as the size tends to infinity. For the standard case of a critical Galton-Watson tree conditioned to be large the limit is the invariant random sin-tree constructed by Aldous (1991). In the condensation regime, we describe in complete generality the asymptotic local behaviour from a random vertex up to its first ancestor with large degree. Beyond this distinguished ancestor, different behaviour may occur, depending on the branching weights. In a subregime of complete condensation, we obtain convergence toward a novel limit tree, that describes the asymptotic shape of the vicinity of the full path from a random vertex to the root vertex. This includes the case where the offspring distribution follows a power law up to a factor that varies slowly at infinity.