Optimal-path cracks in correlated and uncorrelated lattices

The optimal path crack model on uncorrelated surfaces, recently introduced by Andrade et al. [Phys. Rev. Lett. 103, 225503 (2009).], is studied in detail and its main percolation exponents computed. In addition to $\beta /\nu =0.46\pm 0.03$, we report $\gamma /\nu =1.3\pm 0.2$ and $\tau =2.3\pm 0.2$. The analysis is extended to surfaces with spatial long-range power-law correlations, where nonuniversal fractal dimensions are obtained when the degree of correlation is varied. The model is also considered on a three-dimensional lattice, where the main crack is found to be a surface with a fractal dimension of $2.46\pm 0.05$.

Figure 1

Double-logarithmic plot of the mass of the optimal path, $M_{\mathrm{op}}$, in 2D as a function of the linear system size, $L$, for two different types of energy distribution. The lower points ($•$) are measured for uniformly random site energies, being equivalent to weak disorder. The corresponding exponent is $d_{\mathrm{op}}=1.01\pm 0.01$ and the optimal path is a self-affine object [30]. The upper line ($\blacksquare$) shows results obtained for site energies distributed according to a power law, Eq. (1), with ${\beta }_D=1600$, such that the systems are in the strong disorder limit (for the considered system sizes). In the strong disorder limit the optimal path is a self-similar object with a fractal dimension $d_{\mathrm{op}}=1.215\pm 0.005$. The optimal path search was performed using the Dijkstra algorithm [31]. Results have been averaged over ${10}^{10}$ samples for the smallest system size and ${10}^4$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 2

Snapshots of OPC clusters on uncorrelated lattices, generated with the same seed for the uniformly distributed pseudorandom numbers. (a) and (b) show equivalent realizations of the weak disorder limit: In (a) the site energies are distributed according to a power law, Eq. (1), with ${\beta }_D=0.002$, while in (b) they are uniformly distributed. (c) and (d) show snapshots for ${\beta }_D=6$ and ${\beta }_D=100$, respectively, corresponding to the intermediate and the strong disorder regime. The shortest path in the largest optimal path crack cluster is shown in red (medium gray), the sites belonging to the largest cluster but not the shortest path are shown in blue (dark gray), and the sites belonging to other clusters than the largest one are shown in green (light gray). Notice that the shortest path in the largest cluster is independent on the distribution and on the disorder strength. Furthermore, since for the optimal path search only nearest neighbors are considered to be connected, the OPC clusters are connected via nearest and/or next-nearest neighbors. The optimal path crack clusters are trees by construction. For these pictures, the linear system size is $256$.

Figure 3

Double-logarithmic plot of the total mass, $M_{\mathrm{tot}}$ ($\blacksquare$), mass of the largest cluster, $M_{\mathrm{lc}}$ ($•$), and mass of the shortest path, $M_{\mathrm{sp}}$ ($▴$), as a function of the linear system size, $L$, for OPC with uniformly distributed site energies. This energy distribution is equivalent to the weak disorder limit. The three masses scale with the system size according to three different exponents: $d_{\mathrm{tot}}=2.0\pm 0.01$, $d_{\mathrm{lc}}=1.54\pm 0.03$, and $d_{\mathrm{sp}}=1.21\pm 0.02$. For weak disorder, the OPC consists of a largest cluster with dangling ends and many small isolated clusters [see Figs. 2a and 2b]. Results have been averaged over ${10}^8$ samples for the smallest system size and ${10}^2$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 4

Double-logarithmic plot of the total mass, $M_{\mathrm{tot}}$ ($\blacksquare$), mass of the largest cluster, $M_{\mathrm{lc}}$ ($•$), and mass of the shortest path, $M_{\mathrm{sp}}$ ($▴$), as a function of the linear system size, $L$, for site energies distributed according to a power law with disorder parameter ${\beta }_D=6$. The behavior of the total mass and of the mass of the largest cluster reveals the crossover from strong disorder to weak disorder. For small system sizes—equivalent to high disorder—all three masses coincide, while for increasing system size—decreasing disorder—three different exponents emerge, as indicated by the asymptotic dashed lines which have slopes of $2.00$ (for $M_{\mathrm{tot}}$, upper dashed line) and $1.54$ (for $M_{\mathrm{lc}}$, lower dashed line). The solid straight line is a guide to the eye with a slope of $d_{\mathrm{sp}}=1.21\pm 0.02$. For this intermediate value of the disorder there are less isolated clusters than for weak disorder and the largest cluster seems to be dominated by its shortest path [see Fig. 2c]. The inset shows the same masses as the main plot, measured for a higher value of the disorder parameter, ${\beta }_D=30$. For almost the entire range of system sizes the three masses are identical, indicating that these systems are in the strong disorder regime [see Fig. 2d]. The slope of the line gives $d_{\mathrm{sp}}=1.215\pm 0.005$. Results have been averaged over ${10}^8$ samples for the smallest system size and ${10}^2$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 5

Double-logarithmic plot of the cluster size distribution and its second moment for energies uniformly distributed. The main figure shows $N_s={\sum }_{r=s}^{2s-1}{n}_r$ ($s=1,2,4,\dots$) as a function of the cluster size, $s$, for linear system sizes $L=512$ ($\blacksquare$) and $L=1024$ ($•$). The slope of the line gives $1-\tau =-1.3\pm 0.2$. In the inset we see the second moment, $M_2^{\prime }$ ($▴$), of the cluster size distribution as a function of the system size, $L$. The slope of the line gives $\gamma /\nu =1.3\pm 0.2$. Results have been averaged over ${10}^8$ samples for the smallest system size and ${10}^2$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 6

Double-logarithmic plot of the mass of the largest cluster, $M_{\mathrm{lc}}$ $(•)$, for uncorrelated 3D optimal path cracks, as a function of the linear system size, $L$. The slope of the line gives $d_{\mathrm{lc}}=2.46\pm 0.05$. A snapshot of one realization is shown in Fig. 7. Results have been averaged over ${10}^8$ samples for the smallest system size and $50$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 7

Snapshot of a representative configuration of the OPC in 3D, for a system with ${128}^3$ lattice sites. Energies have been assigned to each site in the system according to a truncated power law with ${\beta }_D=100$. In black are the sites where the OPC intercepts the borders of the box.

Figure 8

Density of all removed sites for correlated OPC as a function of the correlation parameter, ${\beta }_C$. For ${\beta }_C=1$, $H=-d/2$, the FBM corresponds to white noise, i.e., the uncorrelated case where the density is independent of the system size. Data for the following system sizes are shown: $L=32$ $(\blacksquare )$, $L=64$ $(•)$, $L=128$ $(▴)$, $L=256$ $(▾)$, $L=512$ $(⧫)$, and $L=1024$ $(*)$. Results have been averaged over $3.2\times {10}^3$ samples for the smallest system size and ${10}^2$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 9

The fractal dimensions $d_{\mathrm{tot}}$ ($\blacksquare$), $d_{\mathrm{lc}}$ ($•$), and $d_{\mathrm{sp}}$ ($▴$) as a function of the correlation parameter, ${\beta }_C$. The plot shows that the long-range spatial correlation changes the exponents of the system. Results have been averaged over $3.2\times {10}^3$ samples for the smallest system size and ${10}^2$ for the largest one. The error bars of the individual points are smaller than the symbols.

Figure 10

The shortest path in the OPC for correlated and uncorrelated energy landscapes. At each lattice site, given by two coordinates $(x,y)$, the third coordinate is the energy at this point. Red regions (mountains) have energy close to unity, while blue regions (valleys) have energy close to zero. The shortest path in the OPC is shown in dark blue (dark gray, left to right) and the first optimal path in dark green (medium gray, front to back). Four different correlation parameters, ${\beta }_C$, have been considered: (a) ${\beta }_C=0$ $(H=-d/2)$ corresponding to the uncorrelated case,- white noise; (b) ${\beta }_C=2.2$ (anticorrelated); (c) ${\beta }_C=3$ $(H=1/2)$ recovering the ordinary Brownian motion where the increments (but not the lattice) are uncorrelated; (d) ${\beta }_C=3.8$, corresponding to strongly correlated lattices. While the crack passes mainly through the mountain tops, the first optimal path passes mainly through the valleys,- hidden in the noisy configurations (a) and (b). For these pictures, the linear system size is 1024.