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

2016/08/02 by Ewain Gwynne, Gwynne, Ewain, Jason Miller +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1608.00954

openalex publication_date 2016/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 (X₁, d₁ , μ₁,η₁) and (X₂, d₂ , μ₂,η₂) are close if they can be isometrically embedded into a common metric space in such a way that the spaces X₁ and X₂ are close in the Hausdorff distance, the measures μ₁ and μ₂ are close in the Prokhorov distance, and the curves η₁ and η₂ are close in the uniform distance.

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