Stochastic PDE Limit of the Six Vertex Model
We study the stochastic six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter Δ → 1 + so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (KPZ) equation. We also pr...
Saved in:
Published in: | Communications in mathematical physics 2020-05, Vol.375 (3), p.1945-2038 |
---|---|
Main Authors: | , , , |
Format: | Article |
Language: | eng |
Subjects: | |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Summary: | We study the
stochastic
six vertex model and prove that under weak asymmetry scaling (i.e., when the parameter
Δ
→
1
+
so as to zoom into the ferroelectric/disordered phase critical point) its height function fluctuations converge to the solution to the Kardar–Parisi–Zhang (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). |
---|---|
ISSN: | 0010-3616 1432-0916 |