We study the stochastic heat equation in two spatial dimensions with a
multiplicative white noise, as the limit of the equation driven by a noise
that is mollified in space and white in time. As the mollification radius
, we
tune the coupling constant near the critical point, and show that the single time
correlation functions converge to a limit written in terms of an explicit nontrivial
semigroup. Our approach consists of two steps. First we show the convergence of the
resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework
of Dimock and Rajeev (J. Phys. A 37:39 (2004), 9157–9173) to our setup of
spatial mollification. Then we match this to the Laplace transform of our
semigroup.