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2018 Uniform infinite half-planar quadrangulations with skewness
Erich Baur, Loïc Richier
Electron. J. Probab. 23: 1-43 (2018). DOI: 10.1214/18-EJP169


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.


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Erich Baur. Loïc Richier. "Uniform infinite half-planar quadrangulations with skewness." Electron. J. Probab. 23 1 - 43, 2018.


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

zbMATH: 1393.05240
MathSciNet: MR3814248
Digital Object Identifier: 10.1214/18-EJP169

Primary: 05C05 , 05C80 , 05C81 , 60F05 , 60J80

Keywords: Boltzmann map , Brownian half-plane , Kesten’s tree , looptree , Multi-type Galton-Watson tree , uniform infinite half-planar quadrangulation

Vol.23 • 2018
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