  We study the limiting behavior of the two-dimensional singular stochastic
stochastic cubic nonlinear complex Ginzburg-Landau with Gibbs measure initial
data. We show that in the appropriate small viscosity and small noise regimes,
the limiting dynamics is given by the deterministic cubic nonlinear
Schr\"odinger equation at Gibbs equilibrium. In order to obtain this
convergence, our approach combines both heat and Schr\"odinger analysis, within
the framework of the Fourier restriction norm method of Bourgain (1993).
