The Annals of Applied Probability

Condensation in critical Cauchy Bienaymé–Galton–Watson trees

Igor Kortchemski and Loïc Richier

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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 $\alpha=1$. 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.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1837-1877.

Dates
Received: May 2018
Revised: November 2018
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566844

Digital Object Identifier
doi:10.1214/18-AAP1447

Mathematical Reviews number (MathSciNet)
MR3914558

Zentralblatt MATH identifier
07057468

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G50: Sums of independent random variables; random walks 60F17: Functional limit theorems; invariance principles 05C05: Trees
Secondary: 05C80: Random graphs [See also 60B20] 60C05: Combinatorial probability

Keywords
Condensation Bienaymé–Galton–Watson tree Cauchy process planar map

Citation

Kortchemski, Igor; Richier, Loïc. Condensation in critical Cauchy Bienaymé–Galton–Watson trees. Ann. Appl. Probab. 29 (2019), no. 3, 1837--1877. doi:10.1214/18-AAP1447. https://projecteuclid.org/euclid.aoap/1550566844


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