Stretched exponential behavior and random walks on diluted hypercubic lattices

TL;DR

This study investigates diffusion on diluted hypercubes at criticality, revealing stretched exponential relaxation with an exponent of 1/3 and exponential growth of relaxation time with dimension, supported by numerical and analytical results.

Contribution

The paper provides large-scale numerical simulations of eigenvalue spectra and relaxation dynamics on diluted hypercubes, confirming theoretical predictions for sparse network models.

Findings

Relaxation functions follow stretched exponentials with exponent 1/3

Relaxation time grows exponentially with hypercube dimension

Eigenvalue spectra align with analytic predictions for sparse networks

Abstract

Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on graphs. In this article we determine numerically through large scale simulations the eigenvalue spectra for this stochastic process and calculate explicitly the time evolution for the autocorrelation function and for the return probability, all at criticality, with hypercube dimensions $N$ up to N=28. We show that at long times both relaxation functions can be described by stretched exponentials with exponent 1/3 and a characteristic relaxation time which grows exponentially with dimension $N$. The numerical eigenvalue spectra are consistent with analytic predictions for a generic sparse network model.