Condensation in critical Cauchy Bienaym\'e-Galton-Watson trees

TL;DR

This paper investigates the structure of large critical Bienaymé-Galton-Watson trees with offspring distributions in the domain of attraction of a Cauchy law, revealing a condensation phenomenon with a macroscopic degree vertex.

Contribution

It establishes limit theorems for specific random walks and applies these to demonstrate condensation in critical trees with Cauchy-like offspring distributions, a novel insight in this context.

Findings

Condensation phenomenon with a macroscopic degree vertex in certain trees

Limit theorems for conditioned Cauchy domain of attraction random walks

Application to the geometry of boundary in non-generic planar maps

Abstract

We are interested in the structure of large Bienaym\'e-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index $\alpha=1$. In stark contrast to the case $\alpha \in (1,2]$, we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges. To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called non-generic of parameter $3/2$). This supports the conjecture that faces in Le Gall & Miermont's $3/2$-stable maps are self-avoiding.