I will present a formula giving the Masur-Veech volumes of 'completed' odd strata of quadratic differentials as a sum over stable graphs. This formula generalizes Delecroix-G-Zograf-Zorich formula in the case of principal strata. The coefficients of the formula are in this case intersection numbers of psi classes with the Witten-Kontsevich combinatorial classes. They naturally appear in the count of integer metrics on ribbon graphs with prescribed odd valencies. The study of the possible degenerations of these ribbon graphs allows to express the difference between the volume of the 'completed' stratum and the volume of the stratum as a linear combination of volumes of boundary strata, with explicit rational coefficients. Several conjectures on the large genus asymptotics of volumes or distribution of cylinders follow from this formula. (work in progress with E. Duryev).[-]

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.[-]