Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology

TL;DR

This paper proves that the uniform infinite half-plane quadrangulation (UIHPQ), when scaled appropriately, converges to the Brownian half-plane in a topology that considers distances, measures, and boundary curves simultaneously.

Contribution

It establishes the convergence of UIHPQ with boundary to the Brownian half-plane in the GHPU topology, extending previous results to include boundary curves.

Findings

UIHPQ converges to Brownian half-plane in GHPU topology

The GHPU topology incorporates distances, measures, and boundary curves

Results apply to both general and simple boundary UIHPQ

Abstract

We prove that the uniform infinite half-plane quadrangulation (UIHPQ), with either general or simple boundary, equipped with its graph distance, its natural area measure, and the curve which traces its boundary, converges in the scaling limit to the Brownian half-plane. The topology of convergence is given by the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) metric on curve-decorated metric measure spaces, which is a generalization of the Gromov-Hausdorff metric whereby two such spaces $(X_1, d_1 , \mu_1,\eta_1)$ and $(X_2, d_2 , \mu_2,\eta_2)$ are close if they can be isometrically embedded into a common metric space in such a way that the spaces $X_1$ and $X_2$ are close in the Hausdorff distance, the measures $\mu_1$ and $\mu_2$ are close in the Prokhorov distance, and the curves $\eta_1$ and $\eta_2$ are close in the uniform distance.