Relation between Dynamic Transport Properties and Static Topological Structure for the Lattice-Animal Model of Branched Polymers

A direct connection is proposed between the "dynamic" transport properties and the "static" topological structure for branched polymers in any number $d$ of spatial dimensions. Specifically, the resistivity exponent $\tilde{\zeta }$ is given by $\tilde{\zeta }=\frac{d_f}{d_l}$, where $d_f$ and $d_l$ are the fractal and topological dimensions (the number of sites within path length $l$ of a given site scales as $M\sim l^{d_l}$). To confirm this new result, we carry out extensive exact and Monte Carlo calculations for $d=2, 3, 4, \mathrm{and} 8$.