Elastic Backbone Defines a New Transition in the Percolation Model

The elastic backbone is the set of all shortest paths. We found a new phase transition at $p_{\mathrm{eb}}$ above the classical percolation threshold at which the elastic backbone becomes dense. At this transition in 2D, its fractal dimension is $1.750\pm 0.003$, and one obtains a novel set of critical exponents ${\beta }_{\mathrm{eb}}=0.50\pm 0.02$, ${\gamma }_{\mathrm{eb}}=1.97\pm 0.05$, and ${\nu }_{\mathrm{eb}}=2.00\pm 0.02$, fulfilling consistent critical scaling laws. Interestingly, however, the hyperscaling relation is violated. Using Binder’s cumulant, we determine, with high precision, the critical probabilities $p_{\mathrm{eb}}$ for the triangular and tilted square lattice for site and bond percolation. This transition describes a sudden rigidification as a function of density when stretching a damaged tissue.

Figure 1

Shown in red are typical elastic backbones obtained for site percolation on a tilted square lattice of linear size $L=512$ and calculated with (a) $p=0.6000$ ($p_c<p<p_{\mathrm{eb}}$), (b) $p=0.7055$ ($p=p_{\mathrm{eb}}$), and (c) $p=0.7500$ ($p>p_{\mathrm{eb}}$). Occupied sites that are in the spanning cluster, but do not belong to the elastic backbone, are shown in blue. Other sites are represented in white.

Figure 2

Density $m_L(p)$ of the elastic backbone as a function of the occupation probability $p$ for site percolation on the triangular lattice for different system sizes. (Inset) Determination of the threshold using finite-size scaling.

Figure 3

Binder’s cumulant $U_L(p)$ for different sizes $L$ as a function of the occupation probability $p$ for site percolation on the triangular lattice. (Inset) The dependence of $p_{\mathrm{eb}}(L)$ on $L^{-1}$ for site percolation on the triangular lattice. $p_{\mathrm{eb}}(L)$ (circles) is obtained from the crossing point of $U_L$ and $U_{2L}$ for each pair of successive $L$ values. Here, we consider $L=64$, 128, 256, 512, 1024, 2048, 4096, and $8192$. Extrapolating through the data points (blue line) to the thermodynamic limit, we obtain $p_{\mathrm{eb}}(\infty )=0.7065\pm 0.0004$ (red dashed line).

Figure 4

Logarithmic plot of the mass $M_{\mathrm{eb}}$ of the elastic backbone as a function of the lattice size $L$ for site percolation at $p_{\mathrm{eb}}$ on the tilted square lattice (orange circles) and on the triangular lattice (blue squares). The same is shown for the value of $p=0.63$, between $p_c$ and $p_{\mathrm{eb}}$, on the tilted square lattice for site percolation (red stars) and for bond percolation (green triangles), with $L$ ranging from 64 to 16384 sites. At $p=p_{\mathrm{eb}}$, the least-squares fit to the data of a power law, $M_{\mathrm{eb}}\sim L^{d_f}$, gives the exponent $d_f=1.7500\pm 0.0003$ for the tilted square lattice (black line) and $d_f=1.7500\pm 0.0002$ for the triangular lattice (blue line). At $p=0.63$ on the tilted square lattice, the least-squares fit to the data of a power law, $M_{\mathrm{eb}}\sim L^d$, gives the exponent $d=1.0000\pm 0.0002$ for site percolation (red line) and $d=1.0000\pm 0.0001$ for bond percolation (green line). In all cases, the errors are smaller than the symbols.

Figure 5

Logarithmic plot of the susceptibility ${\chi }_L$ of the elastic backbone as a function of the lattice size $L$ for site percolation at $p_{\mathrm{eb}}$, with $L$ ranging from 64 to 16384 sites. The least-squares fit to the data of a power law, $\chi (p_{\mathrm{eb}})\sim L^{\gamma /\nu }$, gives the exponent $\gamma /\nu =1.00\pm 0.01$ for the tilted square lattice (orange circles) and $\gamma /\nu =1.00\pm 0.02$ for the triangular lattice (blue squares).

Figure 6

(a) Results from the finite-size scaling analysis of the order parameter $m_L$ obtained for site percolation on the triangular lattice. The growing and decaying curves correspond to values above and below $p_{\mathrm{eb}}$, respectively. The dashed line is the least-squares fit to data in the scaling region above $p_{\mathrm{eb}}$ for Eq. (2). (b) The same as in (a), but for the susceptibility ${\chi }_L$. Blue symbols for $p<p_{\mathrm{eb}}$ and red symbols for $p>p_{\mathrm{eb}}$. Here, the dashed line has slope 1.97. (Inset) Log-log plot of the maximum of ${\chi }_L$ as a function of system size.

Figure 7

(a) Logarithmic plot of the mass $M_{\mathrm{eb}}$ of the elastic backbone as a function of the lattice size $L$ for site percolation at $p_{\mathrm{eb}}(q)$ for two different values of $q$. The least-squares fit to the data of a power law, $M_{\mathrm{eb}}[p_{\mathrm{eb}}(q)]\sim L^{d_f}$, gives the exponent $d_f=1.750\pm 0.005$ for $q=0.06$ (orange circles) and $d_f=1.750\pm 0.003$ for $q=0.80$ (red squares). The results show that the fractal dimension is within error bars the same for both cases. (b) Phase diagram between percolation probability $p$ and the density of diagonals $q$. Three phases can be identified. The nonpercolating phase, where the spanning cluster is absent, bounded on top by the curve $p_c(q)$ (filled circles), which represents the classical percolation thresholds. The nondense phase is bounded by the curve $p_c(q)$ on the bottom and the curve $p_{\mathrm{eb}}(q)$ on the top (stars), which is the critical line along which the order parameter $m_L(p)$ vanishes and the elastic backbone is fractal. Finally, in the dense phase on the top, the elastic backbone becomes dense.