Classification of scaling limits of uniform quadrangulations with a boundary

Understand

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.