[Paper Review] Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes

Kevin Yang|arXiv (Cornell University)|Feb 12, 2020

Stochastic processes and statistical mechanics16 references4 citations

TL;DR

This paper establishes the convergence of the height function in long-range exclusion processes with arbitrary jump lengths to the solution of the Kardar-Parisi-Zhang (KPZ) stochastic PDE under scaling limits, using an approximate microscopic Cole-Hopf transform and local equilibrium techniques. It extends prior results to non-integrable, non-stationary systems, proving weak KPZ universality beyond solvable models.

ABSTRACT

We prove here that the height function associated to non-simple exclusion processes with arbitrary jump-length converges to the solution of the Kardar-Parisi-Zhang SPDE under suitable scaling and renormalization. This extends the work of Dembo-Tsai'16 for arbitrary jump-length and Goncalves-Jara '17 for the non-stationary regime. Thus we answer a ``Big Picture Question" from the AIM workshop on KPZ and also expand on the almost empty set of non-integrable and non-stationary particle systems for which weak KPZ universality is proven. We use an approximate microscopic Cole-Hopf transform as in Dembo-Tsai'16 but we develop tools to analyze local statistics of the particle system via local equilibrium and work of Goncalves-Jara '17. Local equilibrium is done via the one-block step in Guo-Papanicolaou-Varadhan '88 for path-space/dynamic statistics.

Motivation & Objective

• To establish weak KPZ universality for non-integrable, non-stationary exclusion processes with arbitrary jump lengths.
• To extend previous results on KPZ convergence in simple exclusion processes to general long-range dynamics.
• To address the open problem of KPZ universality in discrete, non-Gaussian particle systems lacking exact duality.
• To develop analytical tools for local statistics in non-stationary, non-integrable systems using dynamic one-block estimates.

Proposed method

• Uses an approximate microscopic Cole-Hopf transform to map the height function to a stochastic heat equation (SHE) at the microscopic level.
• Applies dynamic one-block analysis to establish local equilibrium for path-space statistics in non-stationary exclusion processes.
• Employs a torus-based compactification scheme on scale $\mathbb{T}_N = \llbracket -N^{5/4+\varepsilon}, N^{5/4+\varepsilon} \rrbracket$ to control spatial fluctuations.
• Implements time-regularity estimates for the stochastic evolution, leveraging martingale inequalities and heat operator bounds.
• Introduces a macroscopic cutoff function $\chi$ to localize analysis and manage singularities in space-time integrals.
• Uses discrete gradient and Laplacian operators scaled by $N$ to approximate continuum SPDEs, with $\nabla_k^! = N\nabla_k$, $\Delta_k^{!!} = N^2\Delta_k$.

Experimental results

Research questions

• RQ1 Can the KPZ equation emerge as a scaling limit in non-integrable, long-range exclusion processes with arbitrary jump lengths?
• RQ2 How can local equilibrium be established in non-stationary, non-solvable particle systems to enable universality proofs?
• RQ3 What tools are needed to analyze local statistics in the absence of exact duality or integrability?
• RQ4 Can the approximate microscopic Cole-Hopf transform be rigorously justified in non-solvable settings to avoid direct analysis of the singular KPZ equation?
• RQ5 What is the role of time-regularity and compactification in controlling the convergence to the KPZ SPDE?

Key findings

• The height function of long-range exclusion processes with arbitrary jump lengths converges to the Cole-Hopf solution of the KPZ equation under appropriate scaling and renormalization.
• The convergence holds for non-stationary, non-integrable systems, extending previous results limited to nearest-neighbor or solvable models.
• The proof relies on a dynamic one-block analysis that establishes local equilibrium for path-space statistics, even without exact duality.
• The approximate microscopic Cole-Hopf transform enables control of the height function without directly solving the singular KPZ SPDE.
• Time-regularity estimates and compactification on a torus of size $N^{5/4+\varepsilon}$ ensure convergence in distribution to the KPZ solution.
• The method avoids reliance on regularity structures or explicit Gaussian noise, making it applicable to discrete, non-Gaussian particle systems.