[Paper Review] ASEP(q,j) converges to the KPZ equation

Ivan Corwin, Hao Shen+1 more|arXiv (Cornell University)|Feb 5, 2016

Random Matrices and Applications26 references16 citations

TL;DR

This paper establishes that the generalized Asymmetric Exclusion Process ASEP(q,j) converges to the Cole-Hopf solution of the KPZ equation under weak asymmetry scaling. Using a scaling limit approach, the authors prove that the height function of ASEP(q,j) converges to the solution of the KPZ equation, confirming a long-standing conjecture in non-equilibrium statistical mechanics and linking integrable particle systems to stochastic PDEs.

ABSTRACT

We show that a generalized Asymmetric Exclusion Process called ASEP(q,j) introduced by Carinci, Giardina, Redig and Sasamoto converges to the Cole-Hopf solution to the KPZ equation under weak asymmetry scaling.

Motivation & Objective

• To establish the convergence of the generalized ASEP(q,j) process to the KPZ equation under weak asymmetry scaling.
• To bridge integrable stochastic particle systems with the KPZ equation, a fundamental model in non-equilibrium statistical mechanics.
• To confirm the universality of the KPZ equation as the scaling limit of ASEP(q,j), extending previous results on simpler exclusion processes.
• To provide a rigorous mathematical framework for the emergence of the KPZ equation from microscopic stochastic dynamics.

Proposed method

• Applying weak asymmetry scaling to the ASEP(q,j) process, where the asymmetry parameter is tuned to vanish as the system size grows.
• Using the Cole-Hopf transformation to relate the height function of ASEP(q,j) to a solution of the stochastic heat equation.
• Employing martingale methods and tightness arguments to prove convergence of the height function to a solution of the KPZ equation.
• Analyzing the generator of ASEP(q,j) and deriving its hydrodynamic limit under the weakly asymmetric scaling regime.
• Leveraging known results on the convergence of ASEP to the KPZ equation in simpler cases to extend the proof to the generalized ASEP(q,j) model.
• Establishing moment bounds and convergence in law for the height fluctuations using tools from stochastic analysis and integrable probability.

Experimental results

Research questions

• RQ1 Does the generalized ASEP(q,j) process converge to the KPZ equation under weak asymmetry scaling?
• RQ2 What is the limiting behavior of the height function of ASEP(q,j) as the system size increases under weak asymmetry?
• RQ3 How does the Cole-Hopf solution of the KPZ equation emerge from the microscopic dynamics of ASEP(q,j)?
• RQ4 Can the convergence to the KPZ equation be established for the full class of ASEP(q,j) processes, including those with general j and q?

Key findings

• The height function of ASEP(q,j) converges in law to the Cole-Hopf solution of the KPZ equation under weak asymmetry scaling.
• The convergence holds for general parameters q and j, extending previous results to the full class of ASEP(q,j) processes.
• The limiting stochastic PDE is the KPZ equation with space-time white noise, confirming its universality in this class of systems.
• The proof relies on tightness of height fluctuations and convergence of the associated martingale problem to the KPZ equation.