Condensation in critical Cauchy Bienaymé-Galton-Watson trees

We are interested in the structure of large Bienaymé-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α = 1. In stark contrast to the case α ∈ (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.