We study ASEP in a spatially inhomogeneous environment on a torus T(N) = Z/NZ of N sites. A given inhomogeneity ã(x) ∈ (0, ∞), x ∈ T(N), perturbs the overall asymmetric jumping rates r < ℓ ∈ (0, 1) at bonds, so that particles jump from site x to x + 1 with rate rã(x) and from x + 1 to x with rate ℓã(x) (subject to the exclusion rule in both cases). Under the limit N → ∞, we suitably tune the asymmetry (ℓ − r) to zero like N− 21 and the inhomogeneity ã 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 ã(x), which, in particular, includes i.i.d., fractional-Brownian-motion like, and periodic inhomogeneities. The key technical com-ponent of our analysis consists of a host of estimates on the kernel of the semigroup Q(t):= etH for a Hill-type operator H:=12∂xx+A′ (x), and its discrete analog, where A (and its discrete analog) is a generic Hölder continuous function.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
- Inhomogeneous enviornments
- Interacting particle systems
- Stochastic partial differential equations