Continuous-time directed polymers to the Critical SHF via moments

We consider a class of two-dimensional continuous-time directed random polymers in a correlated Brownian environment. The partition functions of these polymers form a discrete approximation of the Critical Stochastic Heat Flow (SHF), constructed by Caravenna, Sun and Zygouras (2023). Using a recent moment based axiomatic characterization of the SHF by Tsai (2024), we prove that the point-to-point partition functions, viewed as random measures, converge to the critical SHF. The proof proceeds by first establishing convergence of all moments through the resolvent method introduced by Gu, Quastel and Tsai (2021) and based on Rajeev (1999) and Dimock and Rajeev (2004). This is followed by establishing tightness and verifying the remaining axioms in the characterization.