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.