Séminaire de Mathématique

Random Square-Tiled Surfaces of Large Genus and Random Multicurves on Surfaces of Large Genus

by Anton Zorich (Université Paris Cité)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

Probability and analysis informal seminar

​(joint work with V. Delecroix, E. Goujard and P. Zograf)

I will remind how Maxim Kontsevich and Paul Norbury have counted metric ribbon graphs and how Maryam Mirzakhani has counted simple closed geodesic multicurves on hyperbolic surfaces. Both counts use Witten-Kontsevich correlators (they will be defined in the lecture with no appeals to quantum gravity).

I will present a formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula, performed by Amol Aggarwal, and the uniform large genus asymptotics of intersection numbers of Witten-Kontsevich correlators, proved by Aggarwal, combined with the results of Kontsevich, Norbury and Mirzakhani, allowed us to describe the structure of a random multi-geodesic on a hyperbolic surface of large genus and of a random square-tiled surface of large genus.

​ As an application I will count oriented meanders on surfaces of any genus and an asymptotic probability to get a meander by a random identification of endpoints of a random braid on a two-component surface of any genus.



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Organized by

Thierry Bodineau, Pieter Lammers, Yilin Wang