Random punctured hyperbolic surfaces & the Brownian sphere


The moduli space of genus-0 hyperbolic surfaces with n punctures comes naturally equipped with a probability measure arising from the Weil-Petersson volume. I will report on preliminary results in collaboration with Nicolas Curien showing that the corresponding random metric space converges after rescaling by $n^{-¼}$ to a multiple of the Brownian sphere as $n \to \infty$ in the Gromov-Prokhorov topology. Removing a small horocyclic neighborhood around each puncture, the same limit is obtained in the Gromov-Hausdorff sense. The proofs rely on an almost-bijection between punctured genus-0 hyperbolic surfaces and certain decorated trees.

Sorbonne Université, Paris, France