  We prove that the stochastic Burgers equation, which is related to the
Kardar-Parisi-Zhang/KPZ equation via weak derivative, is a "critical" scaling
limit for density fluctuations for a family of non-integrable and
non-stationary interacting particle systems. The models we consider cannot be
linearized by a microscopic Cole-Hopf transform or studied directly by the
existing energy solution theory of Goncalves-Jara '14. We develop a novel
method based on comparison to stationary models, a technique that has not yet
been applied to universality of the KPZ equation and nontrivially expands the
set of models for which universality is confirmed. We also study crossover
fluctuations and prove, for the first time, the full transition and phase
diagram from Gaussian to KPZ fluctuations for non-stationary interacting
particle systems, which has not even been done yet for integrable models. We
emphasize the method developed herein applies to a general class of
models/particle systems, but we restrict to a class of zero-range systems whose
non-stationary versions have received widespread interest but had not been
treated in the context of KPZ until now as well as class of non-simple
exclusion processes that we comment on.
