Electronic Journal of Probability

Uniform infinite half-planar quadrangulations with skewness

Erich Baur and Loïc Richier

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We introduce a one-parameter family of random infinite quadrangulations of the half-plane, which we call the uniform infinite half-planar quadrangulations with skewness ($\mathsf{UIHPQ} _p$ for short, with $p\in [0,1/2]$ measuring the skewness). They interpolate between Kesten’s tree corresponding to $p=0$ and the usual $\mathsf{UIHPQ} $ with a general boundary corresponding to $p=1/2$. As we make precise, these models arise as local limits of uniform quadrangulations with a boundary when their volume and perimeter grow in a properly fine-tuned way, and they represent all local limits of (sub)critical Boltzmann quadrangulations whose perimeter tend to infinity. Our main result shows that the family $(\mathsf{UIHPQ} _p)_p$ approximates the Brownian half-planes $\mathsf{BHP} _\theta $, $\theta \geq 0$, recently introduced in [8]. For $p<1/2$, we give a description of the $\mathsf{UIHPQ} _p$ in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 54, 43 pp.

Received: 26 December 2016
Accepted: 18 April 2018
First available in Project Euclid: 7 June 2018

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs 05C05: Trees 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems

uniform infinite half-planar quadrangulation Brownian half-plane Kesten’s tree multi-type Galton-Watson tree looptree Boltzmann map

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Baur, Erich; Richier, Loïc. Uniform infinite half-planar quadrangulations with skewness. Electron. J. Probab. 23 (2018), paper no. 54, 43 pp. doi:10.1214/18-EJP169. https://projecteuclid.org/euclid.ejp/1528358488

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