Markov chain analysis of random walks on disordered medium

TL;DR

This paper investigates the diffusion properties of particles in disordered media modeled by percolation clusters, introducing a new spectral analysis technique to determine dynamical exponents across different disorder regimes.

Contribution

It develops a novel method linking spectral properties of the transition matrix to diffusion exponents, applicable from extreme to weak disorder regimes.

Findings

Established a new scaling relation for the second largest eigenvalue.

Provided an efficient method to extract the spectral dimension $d_s$.

Analyzed the spectral properties using Arnoldi-Saad algorithm.

Abstract

We study the dynamical exponents $d_{w}$ and $d_{s}$ for a particle diffusing in a disordered medium (modeled by a percolation cluster), from the regime of extreme disorder (i.e., when the percolation cluster is a fractal at $p=p_{c}$) to the Lorentz gas regime when the cluster has weak disorder at $p>p_{c}$ and the leading behavior is standard diffusion. A new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used, and the latter is numerically analyzed using the Arnoldi-Saad algorithm. We also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster, $|\ln{\lambda}_{max}|\sim S^{-d_w/d_f}$, which provides a very efficient and accurate method of extracting the spectral dimension $d_s$ where $d_s=2d_f/d_w$.