Superdiffusive Invariance Principle for the two-dimensional Asymmetric Simple Exclusion Process
Abstract
We consider the Asymmetric Simple Exclusion Process (ASEP) at its critical dimension d=2, started from the Bernoulli product measure with density ρ∈ (0,1). We first derive the sharp log(t)2/3 divergence of the bulk diffusivity, predicted via mode-coupling theory in [H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett., 1985], including the exact prefactor, thereby removing the subleading diverging multiplicative corrections in the Tauberian bounds of [H.-T. Yau, Ann. of Math., 2004]. More importantly, we establish an invariance principle for the density fluctuation field on the corresponding superdiffusive scales. On the time scale τ, we rescale space by √τ log(τ)1/3 in the direction of the drift and by √τ in the transverse direction, and prove functional convergence of the rescaled fluctuation process to the solution of a conservative stochastic heat equation with additive noise. This is the same explicit Gaussian Fixed Point obtained in [G. Cannizzaro, Q. Moulard, F. Toninelli, arXiv:2501.00344, 2025] for the two-dimensional Stochastic Burgers Equation (SBE). We also deduce a superdiffusive central limit theorem for the second-class particle under the same anisotropic rescaling. The result holds for any finite-range jump kernel with nonzero drift and uniaxial asymmetry. This is the first space-time scaling limit result for a nonequilibrium interacting particle system at the critical dimension, where Gaussian fluctuations emerge despite logarithmic corrections to the scaling. The proof uses an approximate fluctuation-dissipation relation, which is obtained by combining a sharp quantitative comparison between the ASEP and SBE generators with refined estimates on the SBE Ansatz. An additional key novelty is a tightness argument that overcomes the limitations of the Itô trick on superdiffusive scales.
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BibTeX
@misc{moulard2026superdiffusive,
title = {{Superdiffusive Invariance Principle for the two-dimensional Asymmetric Simple Exclusion Process}},
author = {Quentin Moulard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18847},
eprint = {2610.18847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18847},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Superdiffusive Invariance Principle for the two-dimensional Asymmetric Simple Exclusion Process
AU - Quentin Moulard
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18847
UR - https://arxiv.org/abs/2610.18847
AB - We consider the Asymmetric Simple Exclusion Process (ASEP) at its critical dimension d=2, started from the Bernoulli product measure with density ρ∈ (0,1). We first derive the sharp log(t)^2/3 divergence of the bulk diffusivity, predicted via mode-coupling theory in [H. van Beijeren, R. Kutner, H. Spohn, Phys. Rev. Lett., 1985], including the exact prefactor, thereby removing the subleading diverging multiplicative corrections in the Tauberian bounds of [H.-T. Yau, Ann. of Math., 2004]. More importantly, we establish an invariance principle for the density fluctuation field on the corresponding superdiffusive scales. On the time scale τ, we rescale space by √τ log(τ)^1/3 in the direction of the drift and by √τ in the transverse direction, and prove functional convergence of the rescaled fluctuation process to the solution of a conservative stochastic heat equation with additive noise. This is the same explicit Gaussian Fixed Point obtained in [G. Cannizzaro, Q. Moulard, F. Toninelli, arXiv:2501.00344, 2025] for the two-dimensional Stochastic Burgers Equation (SBE). We also deduce a superdiffusive central limit theorem for the second-class particle under the same anisotropic rescaling. The result holds for any finite-range jump kernel with nonzero drift and uniaxial asymmetry. This is the first space-time scaling limit result for a nonequilibrium interacting particle system at the critical dimension, where Gaussian fluctuations emerge despite logarithmic corrections to the scaling. The proof uses an approximate fluctuation-dissipation relation, which is obtained by combining a sharp quantitative comparison between the ASEP and SBE generators with refined estimates on the SBE Ansatz. An additional key novelty is a tightness argument that overcomes the limitations of the Itô trick on superdiffusive scales.
ER -