Universality of the spectral dimension of percolation clusters

Abstract

We have calculated, via an extensive Monte Carlo simulation, the spectral dimension $\tilde{d}$ of infinite percolation clusters, at different Euclidean dimensions $2<~d<~6$ and $d=\infty$. $\tilde{d}$ is extracted from the asymptotic behavior of the average number $S_N$ of distinct visited sites during an $N$-step random walk. Correction to scaling is shown to be very important. For all $d>~2$, $S_N$ takes the following form: $S_N=a{N}^s\left(\frac{1+b{N}^{-\omega }}{a}\right)$, where $s=\frac{\tilde{d}}{2}$, $0.661<~s<~0.666$, and $\omega \simeq \frac{1}{6}$. We compare our results with existing theories or conjectures relative to the value of $\tilde{d}$.