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We consider a generalized model of random walk in dynamical randomenvironment, and we show that the multiplicative-noise stochastic heat equation(SHE) describes the fluctuations of the quenched density at a certain preciselocation in the tail. The distribution of transition kernels is fixed ratherthan changing under the diffusive rescaling of space-time, i.e., there is nocritical tuning of the model parameters when scaling to the stochastic PDElimit. The proof is done by pushing the methods developed in [arxiv 2304.14279,arXiv 2311.09151] to their maximum, substantially weakening the assumptions andobtaining fairly sharp conditions under which one expects to see the SHE arisein a wide variety of random walk models in random media. In particular we areable to get rid of conditions such as nearest-neighbor interaction as well asspatial independence of quenched transition kernels. Moreover, we observe anentire hierarchy of moderate deviation exponents at which the SHE can be found,confirming a physics prediction of J. Hass.

Invariance principle for the KPZ equation arising in stochastic flows of kernels