Universal fluctuations in the tail probability for d=2 random walks in space-time random environments

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, λ_{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.