Universal Fluctuations in the Tail Probability for d=2 Random Walks in Space-Time Random Environments

TL;DR

This paper investigates the tail probability behavior of two-dimensional random walks in space-time random environments, revealing a universal scaling form characterized by the same coefficient as in one dimension, with a critical fluctuation regime.

Contribution

It extends the understanding of tail probability universality from one-dimensional to two-dimensional RWRE models, identifying a shared coefficient and a critical fluctuation regime.

Findings

Tail probability exhibits a universal scaling form in 2D RWRE.

The same coefficient $mbda_ ext{ext}$ as in 1D models characterizes the tail behavior.

A critical scaling regime for fluctuations at linearly scaled positions in time.

Abstract

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.