Nonuniversality of diffusion exponents in percolation systems

Abstract

We study diffusion on the incipient infinite percolation cluster in $d=2\mathrm{}$ with a power-law distribution of transition rates $P\left(W\right)\mathrm{}\sim W^{-\alpha }$, $\alpha <1\mathrm{}$. Using the exact enumeration method we find that the diffusion exponent ${\overline{d}}_w\left(\alpha \right)$ sticks at its $\alpha =-\infty$ value for $\alpha \leqq 0$. For $\alpha >0\mathrm{}$, ${\overline{d}}_w$ is bounded by $d_f+\frac{1}{\left[{(1-\alpha )}_{\nu }\right]}\leqq {\overline{d}}_w\left(\alpha \right)\leqq {\overline{d}}_w(-\infty )+\frac{\alpha }{[{(1-\alpha )}_{\nu }}]$. Specifically, for small $\alpha$ our numerical results are close to the upper bound, while for larger $\alpha$ they are close to the lower bound.