Structural and dynamical properties of random walk clusters

The authors study the structural and dynamical properties of the clusters generated by a nearest-neighbour random walk embedded in a d-dimensional space. They have focused on the non-trivial case in which the cluster is generated in d=3. The structure of this cluster is characterised by loops for all length scales on the one hand and by the fact that deadends are negligible (upon scaling) on the other hand. The cluster is very dilute and is characterised by fractal dimension d_f=2 and chemical dimension d₁=1.29+or- 0.04. From these results it follows that v identical to d_ld _f approximately=2/3, which is consistent with the formula v=2/d (2<or=d<or=4), obtained using a Flory-type argument. The dynamical diffusion exponents d_w and d^l_w were calculated using the exact enumeration method and found to be d_w=3.45+or-0.10 and d^l_w=2.28+or-0.05. The results suggest that the effect of loops is small but not negligible. They also calculated the fracton dimensionality of the cluster and obtained d_s=1.14+or-0.02. A scaling function is presented for the end-to-end mean square displacement of a random walk performed on a random walk cluster. This scaling function is supported by their numerical results.