Super-Universal Behavior of Outliers Diffusing in a Space-Time Random Environment

I characterize the extreme location and extreme first passage time of a system of $N$ particles independently diffusing in a space-time random environment. I show these extreme statistics are governed by the Kardar-Parisi-Zhang (KPZ) equation and derive their mean and variance. I find the scalings of the statistics depend on the moments of the environment. Each scaling regime forms a universality class which is controlled by the lowest order moment which exhibits random fluctuations. When the first moment is random, the environment plays the role of a random velocity field. When the first moment is fixed but the second moment is random, the environment manifests as fluctuations in the diffusion coefficient. As each higher moment is fixed, the next moment determines the scaling behavior. Since each scaling regime forms a universality class, this model for diffusion forms a super-universality class. I confirm my theoretical predictions using numerics for a wide class of underlying environments.