A conditioning principle for Galton-Watson trees

TL;DR

This paper demonstrates that conditioned infinite Galton-Watson trees converge to a regular tree as the conditioning threshold approaches zero, illustrating a form of entropic repulsion with entropy-free limits.

Contribution

It provides a new limit theorem for conditioned Galton-Watson trees, linking the martingale limit condition to convergence towards a regular tree structure.

Findings

Convergence of conditioned Galton-Watson trees to regular trees as epsilon approaches zero

Illustration of entropic repulsion with entropy-free limits

Identification of the essential minimum of the offspring distribution as a key parameter

Abstract

We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\eps$, converges as $\eps\downarrow 0$ in law to the regular $\mu$-ary tree, where $\mu$ is the essential minimum of the offspring distribution. This gives an example of entropic repulsion where the limit has no entropy.