  We approximate the white-noise driven stochastic heat equation by replacing
the fractional Laplacian by the generator of a discrete time random walk on the
one dimensional lattice, and approximating white noise by a collection of
i.i.d. mean zero random variables. As a consequence, we give an alternative
proof of the weak convergence of the scaled partition function of directed
polymers in the intermediate disorder regime, to the stochastic heat equation;
an advantage of the proof is that it gives the convergence of all moments.
