On the Moments of the (2 + 1) -Dimensional Directed Polymer and Stochastic Heat Equation in the Critical Window

The partition function of the directed polymer model on Z2 + 1 undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochastic heat equation on R2, extending previous work by Bertini and Cancrini (J Phys A Math Gen 31:615, 1998).