Short Talk: Te-Chun Wang

We consider a renormalization of the $d$-dimensional stochastic heat equation (SHE) when the mollification parameter is turned off. Recently, the limiting higher moments of the two-dimensional mollified SHE have been established, and a phase transition is found at $L^{2}$-criticality. By contrast, the above convergences in high dimensions ($d\geq 3$) still remain unknown. To this aim, we will prove this conjecture by showing a completely opposite phenomenon in high dimensions. As an application, we will conclude a precise estimate for the critical exponent of the continuous directed polymer. It is believed that this quantity has a strong connection with the distribution of the limiting partition function of the continuous directed polymer.