Uniform infinite half-planar quadrangulations with skewness

TL;DR

This paper introduces a family of infinite half-plane quadrangulations with a parameter controlling skewness, connecting various models and showing their convergence to Brownian half-planes, with a detailed description for certain skewness levels.

Contribution

It defines a new one-parameter family of models interpolating between known quadrangulations and establishes their convergence to Brownian half-planes, enriching the understanding of geometric limits.

Findings

The models arise as local limits of uniform quadrangulations with boundary.

They approximate Brownian half-planes as the skewness parameter varies.

For p<1/2, the models are described via looptrees from critical Galton-Watson trees.

Abstract

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 (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 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 (UIHPQ$_p$)$_p$ approximates the Brownian half-planes BHP$_\theta$, $\theta\geq 0$, recently introduced in Baur, Miermont, and Ray (2016). For $p<1/2$, we give a description of the UIHPQ$_p$ in terms of a looptree associated to a critical two-type Galton-Watson tree conditioned to survive.