  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).
