Random Weil-Petersson punctured spheres converge, after fourth-root rescaling, to the Brownian sphere.
On the Gromov-Prohorov distance
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We survey some basic results on the Gromov-Prohorov distance between metric measure spaces. (We do not claim any new results.) We give several different definitions and show the equivalence of them. We also show that convergence in the Gromov-Prohorov distance is equivalent to convergence in distribution of the array of distances between finite sets of random points.
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Random punctured hyperbolic surfaces & the Brownian sphere
Random Weil-Petersson punctured spheres converge, after fourth-root rescaling, to the Brownian sphere.