Convergence of the free Boltzmann quadrangulation with simple boundary to the Brownian disk
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Peer-reviewed
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Authors
Gwynne, Ewain
Miller, Jason
Abstract
We prove that the free Boltzmann quadrangulation with simple boundary and fixed perimeter, equipped with its graph metric, natural area measure, and the path which traces its boundary converges in the scaling limit to the free Boltzmann Brownian disk. The topology of convergence is the so-called Gromov-Hausdorff-Prokhorov-uniform (GHPU) topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces. From this we deduce that a random quadrangulation of the sphere decorated by a
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Keywords
Random planar maps, Brownian map, Brownian disk, Quadrangulation with simple boundary, Self-avoiding walk, Gromov-Hausdorff-Prokhorov-uniform topology
Journal Title
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES
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Journal ISSN
0246-0203
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Publisher
Institute of Mathematical Statistics