Appears in collection : Combinatorics, Dynamics and Geometry on Moduli Spaces / Combinatoire, dynamique et géométrie dans les espaces de modules

In this talk we explain results on the large genus asymptotics for intersection numbers between \psiclasses on the moduli space of curves. By combining this result with a combinatorial analysis of formulas of Delecroix-Goujard-Zograf-Zorich, we further describe some features about how random flat surfaces of large genus look. The proof uses a comparison between the recursive relations (Virasoro constraints) that uniquely determine them with the jump probabilities of a certain asymmetric simple random walk.