LAUSR

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathcal{T} \simeq \{0,1,\ldots,N-1\} $ of $ N $ sites. A given inhomogeneity $ \widetilde{\mathsf{a}}(x)\in(0,\infty) $, $ x\in\mathcal{T} $, perturbs the overall asymmetric jumping rates $ r<\ell\in(0,1) $ at bonds, so that particles jump from site $x$ to $x+1$ with rate $r\widetilde{\mathsf{a}}(x)$ and from $x+1$ to $x$ with rate $\ell \widetilde{\mathsf{a}}(x)$ (subject to the exclusion rule in both cases). Under the limit $ N\to\infty $, we suitably tune the asymmetry $ (\ell-r) $ to zero and the inhomogeneity $ \widetilde{\mathsf{a}} $ to unity, so that the two compete on equal footing. At the level of the Gartner (or microscopic Hopf--Cole) transform, we show convergence to a new SPDE --- the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise (or, equivalently, at the level of the height function we show convergence to the Kardar--Parisi--Zhang equation with a mix of spatial and spacetime additive noise). Our method applies to a very general class of inhomogeneity $ \widetilde{\mathsf{a}} $, and in particular includes i.i.d., long-range correlated, and periodic inhomogeneities. The key component of our analysis consists of a host of new estimates on the \emph{kernel} of the semigroup $ \mathcal{Q}:=e^{t\mathcal{Q}} $ for a Hill-type operator $ \mathcal{H}:= \frac12\partial_{xx} + \mathcal{A}'(x) $, and its discrete analog, where $ \mathcal{A} $ (and its discrete analog) is a generic Holder continuous function.