Uniform infinite half-planar quadrangulations with skewness

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 ($\mathsf{UIHPQ} _p$ 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.