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

Large Genus Asymptotics for Intersection Numbers and Principal Strata Volumes of Quadratic Differentials