$t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$

Abstract

We consider finite-range asymmetric exclusion processes on $\mathbb{Z}$ with non-zero drift. The diffusivity $D(t)$ is expected to be of $\mathcal{O}(t^{1/3})$. We prove that $D(t)\ge C{t}^{1/3}$ in the weak (Tauberian) sense that ${\int }_0^{\infty }{e}^{-\lambda t}tD(t)dt\ge C{\lambda }^{-7/3}$ as $\lambda \to 0$. The proof employs the resolvent method to make a direct comparison with the totally asymmetric simple exclusion process, for which the result is a consequence of the scaling limit for the two-point function recently obtained by Ferrari and Spohn. In the nearest neighbor case, we show further that $tD(t)$ is monotone, and hence we can conclude that $D(t)\ge C{t}^{1/3}(\log t{)}^{-7/3}$ in the usual sense.

Version 3. Statement of Theorem 3 is corrected

References in corpus (2)

• Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process
• $(\log t{)}^{2/3}$ law of the two dimensional asymmetric simple exclusion process

Cited by in corpus (24)

• Asymptotics in ASEP with Step Initial Condition
• A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
• Nonlinear fluctuations of weakly asymmetric interacting particle systems
• A Fredholm Determinant Representation in ASEP
• Anisotropic growth of random surfaces in 2+1 dimensions
• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
• A stochastic Burgers equation from a class of microscopic interactions
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• Total Current Fluctuations in ASEP
• Current fluctuations for TASEP: A proof of the Prähofer--Spohn conjecture
• Limit processes for TASEP with shocks and rarefaction fans
• Universality of KPZ equation
• Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers
• Anisotropic KPZ growth in 2+1 dimensions: fluctuations and covariance structure
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Microscopic concavity and fluctuation bounds in a class of deposition processes
• Exact connections between current fluctuations and the second class particle in a class of deposition models
• Superdiffusivity of Asymmetric Energy Model in Dimension One and Two
• Order of current variance and diffusivity in the rate one totally asymmetric zero range process
• Diffusivity of lattice gases
• Current fluctuations of a system of one-dimensional random walks in random environment
• A PDE hierarchy for directed polymers in random environments
• Comments on David Aldous and Persi Diaconis' "Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem"
• Occupation times of long-range exclusion and connections to KPZ class exponents