Condensation in critical Cauchy Bienaymé–Galton–Watson trees

Abstract

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. 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.2010 Mathematics Subject Classification. -Primary 60J80 · 60G50 · 60F17 · 05C05; Secondary 05C80 · 60C05.

Notes on random walks in the Cauchy domain of attraction

“…We are also able to deal with the case p = q = 1/2 when b n /a n → +∞ but we did not state it here for conciseness, since it requires further notations: we refer to Section 7.5 for details, see in particular (7.22). After this article was complete, Kortchemski and Richier [19] proved a similar statement by replacing the ℓ(|b n |) o(1) by some slowly varying function Λ(n), without assuming V1-V2, see [19,Prop. 12].…”

“…Acknowledgements: I am most grateful to Vitali Wachtel for his comments and his suggestions for the improvement of Theorem 3.2, and also to I. Kortchemski and L. Richier for attracting my attention to some subtleties of Theorem 3.4 (and to their article [19]). I thank the referees for their remarks and suggestions, in particular for pointing out references (and an elementary proof) for Theorem 3.7.…”

The boundary of random planar maps via looptrees

“…As suggested in [50,Remark 6.3], we expect the scaling limit to be a multiple of a loop in both cases, although the condensation phenomenon could occur at a scale smaller than the total number of vertices in the critical tree setting. This has been investigated in [37].…”

Critical branching processes in random environment and Cauchy domain of attraction

“…where Λ is a slowly varying function (see the proof of Proposition 12 in Kortchemski and Richier, 2019)…”

“…The authors thank I. Kortchemski for attracting their attention to reference Kortchemski and Richier (2019) which allowed them to weaken the conditions for their main results to hold. C.S.…”