  The partition function of the directed polymer model on Z^{2+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 R^2, 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 R^2, extending previous work by Bertini and Cancrini.
