Many diffusive systems involve correlated random walkers due to a shared environment. Such systems can be modeled as random walks in random environments (RWRE). These models differ from classical diffusion in the behavior of the extremes—the walkers that move the fastest or farthest. In spatial dimension $d=1$, RWRE models have been well studied numerically and analytically and exhibit universal behavior in the Kardar-Parisi-Zhang universality class. Here we study discrete lattice RWRE models in $d=2$. We find that the tail probability exhibits a different universal scaling form, which is nevertheless characterized by the same coefficient, ${\lambda }_{\mathrm{ext}}$, as in the $d=1$ case. We observe a critical scaling regime for fluctuations in the tail probability at positions that scale linearly in time.

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• Brownian motion
• Diffusion
• Random walks
• Kardar–Parisi–Zhang equation
• Stochastic analysis