Uniform infinite half-planar quadrangulations with skewness

We introduce a one-parameter family of random infinite quadrangulations of the halfplane, which we call the uniform infinite half-planar quadrangulations with skewness (UIHPQ(p), for short, with p is an element of [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 >= 0, recently introduced in [8]. 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.