Anomalous diffusion at percolation threshold in high dimensions on 10^18 sites

TL;DR

This paper investigates anomalous diffusion at the percolation threshold in high-dimensional lattices, confirming theoretical predictions about the scaling behavior of random walks in such complex systems.

Contribution

It introduces an efficient method for simulating random walks on large high-dimensional lattices at the percolation threshold.

Findings

Confirmed the expected time-dependence of end-to-end distance in 7D lattices.

Validated corrections to asymptotic behavior in high-dimensional percolation.

Demonstrated the feasibility of simulating lattices with over 10^18 sites.

Abstract

Using an inverse of the standard linear congruential random number generator, large randomly occupied lattices can be visited by a random walker without having to determine the occupation status of every lattice site in advance. In seven dimensions, at the percolation threshold with L^7 sites and L < 420, we confirm the expected time-dependence of the end-to-end distance (including the corrections to the asymptotic behavior).