# Scaling laws for random walks in long-range correlated disordered media

## TL;DR

This paper investigates how long-range correlations in disordered media affect diffusion scaling laws, revealing different diffusion behaviors at criticality depending on correlation strength.

## Contribution

It introduces a detailed analysis of diffusion in correlated disordered media using percolation models and exact enumeration, highlighting the impact of correlation decay on diffusion properties.

## Key findings

- At criticality, sub-diffusive behavior characterized by walk dimension $d_w$ varies with correlation exponent $a$.
- Weak correlations lead to normal diffusion above percolation threshold.
- Strong correlations may result in anomalous diffusion, especially for small $a$.

## Abstract

We study the scaling laws of diffusion in two-dimensional media with long-range correlated disorder through exact enumeration of random walks. The disordered medium is modelled by percolation clusters with correlations decaying with the distance as a power law, $r^{-a}$, generated with the improved Fourier filtering method. To characterize this type of disorder, we determine the percolation threshold $p_{\text c}$ by investigating cluster-wrapping probabilities. At $p_{\text c}$, we estimate the (sub-diffusive) walk dimension $d_{\text w}$ for different correlation exponents $a$. Above $p_{\text c}$, our results suggest a normal random walk behavior for weak correlations, whereas anomalous diffusion cannot be ruled out in the strongly correlated case, i.e., for small $a$.