SPDE Limit of Weakly Inhomogeneous ASEP

Abstract

We study ASEP in a spatially inhomogeneous environment on a torus ${\mathbb{T}}^{(N)}=\mathbb{Z}/N\mathbb{Z}$ of $N$ sites. A given inhomogeneity $\tilde{a}(x)\in (0,\infty )$, $x\in \mathbb{T}$, 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 $r\tilde{a}(x)$ and from $x+1$ to $x$ with rate $\ell \tilde{a}(x)$ (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 $\tilde{a}(x)$ 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 $\tilde{a}(x)$, which, in particular, includes i.i.d., long-range correlated, and periodic inhomogeneities. The key technical component of our analysis consists of a host of estimates on the \emph{kernel} of the semigroup $\mathcal{Q}(t):=e^{t\mathcal{H}}$ for a Hill-type operator $\mathcal{H}:=\frac{1}{2}{\partial }_{xx}+{\mathcal{A}}^{\prime }(x)$, and its discrete analog, where $\mathcal{A}$ (and its discrete analog) is a generic Hölder continuous function.

45 pages, 2 figures