TY - JOUR

T1 - Spde limit of weakly inhomogeneous asep

AU - Corwin, Ivan

AU - Tsai, Li Cheng

N1 - Funding Information:
We thank Yu Gu and Hao Shen for useful discussions during the writing of the first manuscript, and particularly acknowledge Hao Shen for pointing to us the argument in [Lab17, Proof of Proposition 3.8]. Ivan Corwin was partially supported by the Packard Fellowship for Science and Engineering, and by the NSF through DMS-1811143 and DMS-1664650. Li-Cheng Tsai was partially supported by the Simons Foundation through a Junior Fellowship and by the NSF through DMS-1712575.
Publisher Copyright:
© 2020, Institute of Mathematical Statistics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Inhomogeneous enviornments

KW - Interacting particle systems

KW - Stochastic partial differential equations

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U2 - 10.1214/20-EJP565

DO - 10.1214/20-EJP565

M3 - Article

AN - SCOPUS:85098891420

VL - 25

SP - 1

EP - 55

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 156

ER -