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

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.