Classification of scaling limits of uniform quadrangulations with a boundary

Abstract

We study non-compact scaling limits of uniform random planar quadrangulations with a boundary when their size tends to infinity. Depending on the asymptotic behavior of the boundary size and the choice of the scaling factor, we observe different limiting metric spaces. Among well-known objects like the Brownian plane or the infinite continuum random tree, we construct two new one-parameter families of metric spaces that appear as scaling limits: the Brownian half-plane with skewness parameter $\theta$ and the infinite-volume Brownian disk of perimeter $\sigma$. We also obtain various coupling and limit results clarifying the relation between these objects.