Condensation in critical Cauchy Bienaymé–Galton–Watson trees

Abstract

We are interested in the structure of large Bienayme–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 (see Figure 1). 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 nongeneric of parameter $3/2$). This supports the conjecture that faces in Le Gall and Miermont’s $3/2$-stable maps are self-avoiding.