Strong Disorder for Stochastic Heat Flow and 2D Directed Polymers

The critical 2D Stochastic Heat Flow (SHF) is a universal measure-valued process that provides a notion of solution to the ill-defined 2D stochastic heat equation. We investigate the SHF in the large-time and strong-disorder regimes, proving a sharp form of local extinction: we identify the rate at which the distribution collapses to zero. We also identify the spatial scale governing the transition from vanishing mass to diverging mass, and from extinction to an averaged behavior. Corresponding results are established for the partition functions of 2D directed polymers, yielding precise free-energy estimates. Our proof provides a unified framework of change of measure and coarse-graining arguments. These results offer new insights into the 2D stochastic heat equation regularized via space-time discretization: for any regime of supercritical disorder strength $\beta$, including the case where $\beta > 0$ is kept fixed, the solution exhibits fluctuations on a superdiffusive scale.