  We consider a generalized model of random walk in dynamical random
environment, and we show that the multiplicative-noise stochastic heat equation
(SHE) describes the fluctuations of the quenched density at a certain precise
location in the tail. The distribution of transition kernels is fixed rather
than changing under the diffusive rescaling of space-time, i.e., there is no
critical tuning of the model parameters when scaling to the stochastic PDE
limit. The proof is done by pushing the methods developed in [arxiv 2304.14279,
arXiv 2311.09151] to their maximum, substantially weakening the assumptions and
obtaining fairly sharp conditions under which one expects to see the SHE arise
in a wide variety of random walk models in random media. In particular we are
able to get rid of conditions such as nearest-neighbor interaction as well as
spatial independence of quenched transition kernels. Moreover, we observe an
entire hierarchy of moderate deviation exponents at which the SHE can be found,
confirming a physics prediction of J. Hass.
