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We introduce a one-parameter family of random infinite quadrangulations ofthe half-plane, which we call the uniform infinite half-planar quadrangulationswith 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 usualUIHPQ with a general boundary corresponding to $p=1/2$. As we make precise,these models arise as local limits of uniform quadrangulations with a boundarywhen their volume and perimeter grow in a properly fine-tuned way, and theyrepresent all local limits of (sub)critical Boltzmann quadrangulations whoseperimeter tend to infinity. Our main result shows that the family(UIHPQ$_p$)$_p$ approximates the Brownian half-planes BHP$_\theta$, $\theta\geq0$, recently introduced in Baur, Miermont, and Ray (2016). For $p<1/2$, we givea description of the UIHPQ$_p$ in terms of a looptree associated to a criticaltwo-type Galton-Watson tree conditioned to survive.

Uniform infinite half-planar quadrangulations with skewness