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June 2019 Condensation in critical Cauchy Bienaymé–Galton–Watson trees
Igor Kortchemski, Loïc Richier
Ann. Appl. Probab. 29(3): 1837-1877 (June 2019). DOI: 10.1214/18-AAP1447

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.

Citation

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Igor Kortchemski. Loïc Richier. "Condensation in critical Cauchy Bienaymé–Galton–Watson trees." Ann. Appl. Probab. 29 (3) 1837 - 1877, June 2019. https://doi.org/10.1214/18-AAP1447

Information

Received: 1 May 2018; Revised: 1 November 2018; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057468
MathSciNet: MR3914558
Digital Object Identifier: 10.1214/18-AAP1447

Subjects:
Primary: 05C05, 60F17, 60G50, 60J80
Secondary: 05C80, 60C05

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 3 • June 2019
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