Random Weil-Petersson punctured spheres converge, after fourth-root rescaling, to the Brownian sphere.
Moduli spaces of hyperbolic surfaces and their Weil-Petersson volumes
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abstract
Moduli spaces of hyperbolic surfaces may be endowed with a symplectic structure via the Weil-Petersson form. Mirzakhani proved that Weil-Petersson volumes exhibit polynomial behaviour and that their coefficients store intersection numbers on moduli spaces of curves. In this survey article, we discuss these results as well as some consequences and applications.
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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.