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

Abstract

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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}$ is given by $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\zeta}}=\frac{{d}_{f}}{{d}_{l}}$, 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 $M\ensuremath{\sim}{l}^{{d}_{l}}$). To confirm this new result, we carry out extensive exact and Monte Carlo calculations for $d=2, 3, 4, \mathrm{and} 8$.