Two dimensional random walks in random environments exhibit 2+1D KPZ fluctuations despite the 2+1D KPZ not being well defined

The Kardar-Parisi-Zhang (KPZ) equation is a well defined stochastic partial differential equation in 1 spatial and 1 time dimension (1+1D). Its extension to 2+1D seems trivial at first glance, however, the stochastic term and non-linearity in 2 spatial dimensions make the equation intractable. Nevertheless, just as the 1+1D KPZ equation defines a universality class, there exists a class of (discrete) models in 2D which exhibit universal behavior that is conjectured to be governed by this 2+1D KPZ. The limiting distribution and statistics of this universality class are highly sensitive to how the noise is mollified (i.e. averaged in space). Previously, we have shown that the Random Walks in Random Environments (RWRE) model for diffusion in 1+1D displays fluctuations characterized by the 1+1D KPZ universality class. Here, we study the RWRE in 2+1D using analytics and numerics to probe the characteristics of the 2+1D KPZ universality class. We show that KPZ fluctuations occur in a scaling regime controlled by the underlying statistics of the environment. We determine if these fluctuations display a crossover from a weak disorder regime, where the fluctuations are Gaussian, to a strong disorder regime, where the fluctuations are non-Gaussian.