  In this paper we analyze the large genus asymptotics for intersection numbers
between $\psi$-classes, also called correlators, on the moduli space of stable
curves. Our proofs proceed through a combinatorial analysis of the recursive
relations (Virasoro constraints) that uniquely determine these correlators,
together with a comparison between the coefficients in these relations with the
jump probabilities of a certain asymmetric simple random walk. As an
application of this result, we provide the large genus limits for Masur-Veech
volumes and area Siegel-Veech constants associated with principal strata in the
moduli space of quadratic differentials. These confirm predictions of
Delecroix-Goujard-Zograf-Zorich from 2019.
