Self-affinity in the gradient percolation problem

We study the scaling properties of the solid-on-solid front of the infinite cluster in two-dimensional gradient percolation. We show that such an object is self-affine with a Hurst exponent equal to $2∕3$ up to a cutoff length $\sim g^{-4∕7}$, where $g$ is the gradient. Beyond this length scale, the front position has the character of uncorrelated noise. Importantly, the self-affine behavior is robust even after removing local jumps of the front. The previously observed multiaffinity is due to the dominance of overhangs at small distances in the structure function. This is a crossover effect.

Figure 1

Top-side and bottom-side SOS fronts based on the perimeter of the cluster connected to the $p=1$ edge (shown in the inset as dots). We also show the filtered $j_0\left(i\right)$ front [see Eq. (6)].

Figure 2

Data collapse of the averaged wavelet coefficients for the bottom side front based on lattice size $L_j=64$ to 8192, while $L_i=2048$. We show that $g=1∕L_j$. The straight line has a slope of $\zeta +1∕2$, [see Eq. (2)].

Figure 3

Data collapse of the averaged wavelet coefficients for the smoothed $j_0\left(i\right)$ based on the bottom side front. The lattice sizes and gradients are as in Fig. 2. The long-dashed line has a slope of 7/6 [see Eq. (2)], whereas the dotted line has a slope of $1=1∕2+1∕2$, consistent with uncorrelated random walks.

Figure 4

$C_k(n,g)$ as a function of $n$ for $k=1$, 2, and 3. The three leftmost straight lines have slopes according to Eq. (8), while the bold middle line has a slope equal to $2∕3$ in accordance with Eq. (9). For large $n$ the structure functions become flat indicating that one has reached the decorrelated regime.