| Résume||It is common in mathematics to study decompositions of compoundobjects into primitive blocks. For example, the Erdos-Kac Theorem
describes the prime decomposition of a random integer number into prime factors. The Theorem of Goncharov describes the decomposition
of a random permutation into disjoint cycles.
I will present our 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 Aggarwal and the uniform large genus asymptotics of intersection numbers of psi-classes on the moduli spaces of complex curves proved by Aggarwal allowed us to describe the decomposition of a random square-tiled surface of large genus into maximal horizontal cylinders. Our results imply, in particular, that with a probability which tends to 1, as genus grows, all ``corners'' of a random square-tiled surface live on the same horizontal and on the same vertical critical leave, and with probability 71% a random square-tiled surface is composed of a single horizontal band of squares.
(joint work with V. Delecroix, E. Goujard and P. Zograf)|