Uniform infinite half-planar quadrangulations with skewness

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 $p\in [0,1\mathrm{/}2]$ measuring the skewness). They interpolate between Kesten’s tree corresponding to $p=0$ and the usual UIHPQ with a general boundary corresponding to $p=1\mathrm{/}2$. 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${}_{\theta }$, $\theta\geq 0$,recentlyintroducedinBaur,Miermont,andRay(2016)\mathrm{.}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.