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Planarity and non-separating cycles in uniform high genus quadrangulations
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We study large uniform random quadrangulations whose genus grow linearly with the number of faces, whose local convergence was recently established by Budzinski and the author arXiv:1902.00492,arXiv:2012.05813. Here we study several properties of these objects which are not captured by the local topology. Namely we show that balls around the root are planar whp up to logarithmic radius, and we prove that there exists short non-contractible cycles with positive probability.
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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.
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