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

Area of Science:

• Physics
• Statistical Mechanics
• Complex Systems

Background:

• Diffusive systems often feature correlated random walkers due to shared environments.
• Random walks in random environments (RWRE) model these complex diffusive behaviors.
• RWRE exhibit distinct behaviors in extreme outcomes compared to classical diffusion.

Purpose of the Study:

• Investigate discrete lattice random walks in random environments (RWRE) in two spatial dimensions (d=2).
• Characterize the universal scaling form of tail probability for extreme walkers in d=2.
• Compare findings with established one-dimensional (d=1) RWRE models.

Main Methods:

• Analysis of discrete lattice models in d=2.
• Numerical and analytical studies of random walks in random environments.
• Examination of tail probability distributions and scaling behaviors.

Main Results:

• The tail probability in d=2 RWRE exhibits a novel universal scaling form.
• This scaling form is characterized by the same external coefficient (λ_ext) as observed in d=1.
• A critical scaling regime for tail probability fluctuations was identified, scaling linearly with time.

Conclusions:

• Two-dimensional RWRE models display distinct universal scaling properties for extreme walkers.
• The findings extend the understanding of universality in random walk models to higher dimensions.
• The study highlights the importance of environmental correlations in diffusive processes.