  We consider directed polymer models involving multiple non-intersecting
random walks moving through a space-time disordered environment in one spatial
dimension. For a single random walk, Alberts, Khanin and Quastel proved that
under intermediate disorder scaling (in which time and space are scaled
diffusively, and the strength of the environment is scaled to zero in a
critical manner) the polymer partition function converges to the solution to
the stochastic heat equation with multiplicative white noise. In this paper we
prove the analogous result for multiple non-intersecting random walks started
and ended grouped together. The limiting object now is the multi-layer
extension of the stochastic heat equation introduced by O'Connell and Warren.
