Asymptotic Scaling Behavior of Self-Avoiding Walks on Critical Percolation Clusters

We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters’ fractal nature. We enumerate walks of over ${10}^4$ steps, far more than has ever been possible. The scaling exponent $\nu$ for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfectly fits our data.

Figure 1

Backbone of a critical percolation cluster on a cubic lattice of ${300}^3$ sites. The two ends are connected via periodic boundary conditions; coloring indicates shortest-path distance to the origin.

Figure 2

Schematic picture of a tree hierarchy of nested blobs. The starting position for the walks is marked black. Walk segments through the blobs are generated in the order $E\to A$.

Figure 3

Mean squared end-to-end distance vs number of steps for SAWs on incipient critical clusters (red) and backbones (green) on a log-log scale. The lines show the results from different least-squares fits. $f_1$: Eq. (2) with $N=800–12800$ (ic) and $N=1131–12800$ (bb); $f_2$ (inset): Eq. (2), $N=13–100$; $g$: Eq. (3), $N=25–12800$. The factor $N^{-1.33}$ ($\approx N^{-2\nu }$) serves to magnify the differences.

Figure 4

Measured probability densities of the entropy $\ln Z$ for various lengths. The dashed lines are normal distributions with the same mean values and variances (no fit involved).

Figure 5

Mean entropy (red triangles), logarithm of the average number of conformations (green squares), and lognormal approximation (blue diamonds) vs $N$ on a log-linear scale. The estimates for $\ln [Z]$ are biased for $N>200$ due to large deviations. The inset shows fits of $[\ln Z]$ using Eq. (7) (dotted) and Eq. (10) (continuous), respectively.