Stochastic PDE Limit of the Six Vertex Model

TL;DR

This paper proves that the height function fluctuations of the stochastic six vertex model under weak asymmetry scaling converge to the KPZ equation, establishing a connection between a lattice model and a fundamental stochastic PDE.

Contribution

It introduces an exact microscopic Hopf-Cole transform for the stochastic six vertex model and uses Markov duality to prove convergence to the KPZ equation.

Findings

Height function fluctuations converge to KPZ equation.

Stochastic Gibbs states converge to stochastic Burgers equation.

Self-averaging is established via Bethe ansatz contour integrals.

Abstract

We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter $\Delta\to 1^+$ so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the KPZ equation. We also prove that the one-dimensional family of stochastic Gibbs states for the symmetric six vertex model converge under the same scaling to the stationary solution to the stochastic Burgers equation. Our proofs rely upon the Markov (self) duality of our model. The starting point is an exact microscopic Hopf-Cole transform for the stochastic six vertex model which follows from the model's known one-particle Markov self-duality. Given this transform, the crucial step is to establish self-averaging for specific quadratic function of the transformed height function. We use the model's two-particle self-duality to produce explicit expressions (as Bethe ansatz contour integrals) for conditional expectations from which we extract time-decorrelation and hence self-averaging in time. The crux of our Markov duality method is that the entire convergence result reduces to precise estimates on the one-particle and two-particle transition probabilities. Previous to our work, Markov dualities had only been used to prove convergence of particle systems to linear Gaussian SPDEs (e.g. the stochastic heat equation with additive noise).