Abstract 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 spatial location in the tail called the critical scale . The distribution of transition kernels is fixed rather than changing under the diffusive rescaling of space-time, that is, there is no tuning of the model parameters needed to observe the stochastic PDE limit. The proof is done by pushing the methods developed in DDP24a, DDP24b 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 Has25 and mirroring a result from HQ18 in the context of this model.
Shalin Parekh (Thu,) studied this question.