Strong disorder for Stochastic Heat Flow and 2D Directed Polymers

Abstract: The critical 2D Stochastic Heat Flow (SHF) is a universal process of random measures, arising as the scaling limit of two-dimensional directed polymer partition functions (or solutions to the stochastic heat equation with mollified noise) under a critical renormalisation of disorder strength. We investigate the SHF in the strong disorder (or super-critical) regime, proving that it vanishes locally with an optimal doubly-exponential decay rate in the disorder intensity. We also establish a strengthened version of this result for 2D directed polymers, from which we deduce sharp bounds on the free energy. Our proof is based on estimates for truncated and fractional moments, exploiting refined change of measure and coarse-graining techniques.