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

by Anton Zorich (Université Paris Cité)

Friday Apr 21, 2023, 2:00 PM → 4:00 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

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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