Appears in collection : Probability and Geometry in, on and of non-Euclidian spaces / Probabilités et géométrie dans, sur et des espaces non-euclidiens

Square-tiled surfaces are surfaces obtained by gluing euclidean squares along the edge. They naturally inherit a flat metric with conical singularities from the euclidean plane. In this talk we focus on the family of orientable square-tiled surfaces whose sides are identified by translations and half-turns. I will present a formula for the asymptotic count of such square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This formula relies on the results of Kontsevich and Norbury for the count of metric ribbon graphs, and is also related to Mirzakhani's count of simple closed geodesic multicurves on hyperbolic surfaces. Combining this formula with recent results of Aggarwal, we are able to describe the structure of a random square-tiled surface of large genus, but also the structure of a random geodesic multicurve on a hyperbolic surface of large genus. This a joint work with V. Delecroix, A. Zorich and P. Zograf.