The scaling behavior of linear polymers in disordered media modeled by self-avoiding random walks (SAWs) on the backbone of two- and three-dimensional percolation clusters at their critical concentrations $p_{\mathrm{c}}$ is studied. All possible SAW configurations of N steps on a single backbone configuration are enumerated exactly. We find that the moments of order q of the total number of SAWs obtained by averaging over many backbone configurations display multifractal behavior; i.e., different moments are dominated by different subsets of the backbone. This leads to generalized coordination numbers ${\mu }_q$ and enhancement exponents ${\gamma }_q,$ which depend on q. Our numerical results suggest that the relation ${\mu }_1{=p}_{\mathrm{c}}\mu$ between the first moment ${\mu }_1$ and its regular lattice counterpart μ is valid.

• Received 7 January 2000

DOI:https://doi.org/10.1103/PhysRevE.61.6858