Classification of scaling limits of uniform quadrangulations with a boundary

TL;DR

This paper investigates the scaling limits of large uniform quadrangulations with boundaries, revealing new metric space families and clarifying their relationships with known objects like the Brownian plane.

Contribution

It introduces two new one-parameter families of scaling limit spaces for quadrangulations with boundary, expanding the understanding of their asymptotic geometric structures.

Findings

Identification of the Brownian half-plane with skewness parameter as a scaling limit.

Construction of the infinite-volume Brownian disk of perimeter σ.

Clarification of relationships between new and existing limit objects.

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.