Scaling Limit of Random Planar Quadrangulations with a Boundary

TL;DR

This paper studies the scaling limits of large random planar quadrangulations with a boundary, showing convergence to different limiting metric spaces depending on boundary size, including the Brownian map and Aldous's CRT.

Contribution

It establishes the convergence of scaled quadrangulations with boundary to new limiting spaces, extending known results to boundary cases and varying boundary sizes.

Findings

Convergence to a Hausdorff dimension 4 space with boundary for finite boundary size.

Limit is the Brownian map when boundary size tends to zero.

Limit is Aldous's CRT when boundary size tends to infinity.

Abstract

We discuss the scaling limit of large planar quadrangulations with a boundary whose length is of order the square root of the number of faces. We consider a sequence $(\sigma_n)$ of integers such that $\sigma_n/\sqrt{2n}$ tends to some $\sigma\in[0,\infty]$. For every $n \ge 1$, we call $q_n$ a random map uniformly distributed over the set of all rooted planar quadrangulations with a boundary having $n$ faces and $2\sigma_n$ half-edges on the boundary. For $\sigma\in (0,\infty)$, we view $q_n$ as a metric space by endowing its set of vertices with the graph metric, rescaled by $n^{-1/4}$. We show that this metric space converges in distribution, at least along some subsequence, toward a limiting random metric space, in the sense of the Gromov--Hausdorff topology. We show that the limiting metric space is almost surely a space of Hausdorff dimension 4 with a boundary of Hausdorff dimension 2 that is homeomorphic to the two-dimensional disc. For $\sigma=0$, the same convergence holds without extraction and the limit is the so-called Brownian map. For $\sigma=\infty$, the proper scaling becomes $\sigma_n^{-1/2}$ and we obtain a convergence toward Aldous's CRT.