SPDE limit of weakly inhomogeneous ASEP

Abstract

We study ASEP in a spatially inhomogeneous environment on a torus $ \mathbb {T}^{(N)} = \mathbf {Z}/N\mathbf {Z}$ of $ N $ sites. A given inhomogeneity $ \widetilde { \mathbb {a} } (x)\in (0,\infty ) $, $ x\in \mathbb {T}^{(N)} $, 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 $r \widetilde { \mathbb {a} } (x)$ and from $x+1$ to $x$ with rate $\ell \widetilde { \mathbb {a} } (x)$ (subject to the exclusion rule in both cases). Under the limit $ N\to \infty $, we suitably tune the asymmetry $ (\ell -r) $ to zero like $N^{-\frac {1}{2}}$ and the inhomogeneity $ \widetilde { \mathbb {a} } $ to unity, so that the two compete on equal footing. At the level of the Gärtner (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. 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 general class of $ \widetilde { \mathbb {a} } (x) $, which, in particular, includes i.i.d., fractional-Brownian-motion like, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the kernel of the semigroup $ \mathcal {Q} (t):=e^{t \mathcal {H} } $ for a Hill-type operator $ \mathcal {H} := \frac {1}{2}\partial _{xx} + \mathcal {A} '(x) $, and its discrete analog, where $ \mathcal {A} $ (and its discrete analog) is a generic Hölder continuous function.