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- W3116354130 abstract "We study ASEP in a spatially inhomogeneous environment on a torus $ mathbb {T}^{(N)} = mathbf {Z}/Nmathbf {Z}$ of $ N $ sites. A given inhomogeneity $ widetilde { mathbb {a} } (x)in (0,infty ) $, $ xin 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 $ Nto 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." @default.
- W3116354130 created "2021-01-05" @default.
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- W3116354130 date "2020-01-01" @default.
- W3116354130 modified "2023-09-30" @default.
- W3116354130 title "SPDE limit of weakly inhomogeneous ASEP" @default.
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- W3116354130 doi "https://doi.org/10.1214/20-ejp565" @default.
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