Possible Breakdown of the Alexander-Orbach Rule at Low Dimensionalities

Abstract

Simple conditions are presented under which the fractal dimension of a random walk on an aggregate, $d_w$, is given by $d_w=D+1\mathrm{}$, where $D$ is the aggregate's fractal dimension. These conditions are argued (with one simple speculative assumption) to apply for $D<\mathrm{}2\mathrm{}$, implying a breakdown of the Alexander-Orbach rule $d_w=\frac{3D}{2}$. Existing results for percolation clusters, lattice animals, and diffusion-limited aggregates seem to favor our new rule.