Convergence of the free Boltzmann quadrangulation with simple boundary\n to the Brownian disk

2017/01/18 by Ewain Gwynne, Gwynne, Ewain, Jason Miller +1
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1701.05173

openalex publication_date 2017/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the free Boltzmann quadrangulation with simple boundary and\nfixed perimeter, equipped with its graph metric, natural area measure, and the\npath which traces its boundary converges in the scaling limit to the free\nBoltzmann Brownian disk. The topology of convergence is the so-called\nGromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the\nGromov-Hausdorff topology for curve-decorated metric measure spaces. From this\nwe deduce that a random quadrangulation of the sphere decorated by a 2l-step\nself-avoiding loop converges in law in the GHPU topology to the random\ncurve-decorated metric measure space obtained by gluing together two\nindependent Brownian disks along their boundaries.\n

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