Clay Mathematics Institute

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.

We consider directed polymer models involving multiple non-intersecting\nrandom walks moving through a space-time disordered environment in one spatial\ndimension. For a single random walk, Alberts, Khanin and Quastel proved that\nunder intermediate disorder scaling (in which time and space are scaled\ndiffusively, and the strength of the environment is scaled to zero in a\ncritical manner) the polymer partition function converges to the solution to\nthe stochastic heat equation with multiplicative white noise. In this paper we\nprove the analogous result for multiple non-intersecting random walks started\nand ended grouped together. The limiting object now is the multi-layer\nextension of the stochastic heat equation introduced by O'Connell and Warren.\n