Super-Universal Behavior of Outliers Diffusing in (Space-Time) Random Environments*

Random walks have historically been used to model diffusive systems, but require many simplifying assumptions. Recently there has been increasing interest in a new model which accounts for space-time correlations between the particles through a shared environment (the Random Walks in a Random Environment model). This model for diffusion recovers the same bulk properties as classical diffusion (i.e. statistics of a typical particle). However, the tail of the distribution (i.e. particles that have moved the farthest, the fastest) displays fluctuations relating to the Kardar-Parisi-Zhang (KPZ) universality class. This model has been studied in the case of random walks on a 1D lattice where particles can only move to neighboring sites and the transition probabilities at each site and time are drawn from a random distribution. We study a generalization of this model where particles can move to any site on the lattice and show that the first two moments of the tail probability converge to those of the KPZ equation. We translate these results into predictions of physical measurements of a system of diffusing particles – the position of the maximum particle and fastest first passage time – and verify the predictions numerically. We find the scaling exponents demonstrate super-universal behavior as determined by the statistics of the random environment.