Correlation length exponent in the three-dimensional fuse network

We present numerical measurements of the critical correlation length exponent $\nu$ in the three-dimensional fuse model. Using sufficiently broad threshold distributions to ensure that the system is the strong-disorder regime, we determine $\nu$ to be $\nu =0.83\pm 0.04$ based on analyzing the fluctuations of the survival probability. This value is different from that of ordinary percolation, which is 0.88.

Figure 1

Normalized histogram where each bin is averaged over a plane orthogonal to the (1,1,1) direction for $D=10$. The position of the damage zone for each sample is adjusted so that its center of mass is at the center of the plot before averaging. This is in contrast to the damage profile averaging performed in Ref. 18 where the maximum of each damage profile was used as center point. The latter procedure will always produce a cusp.

Figure 2

Plot of natural logarithm of the critical damage fluctuations $W_c={(⟨p^2⟩-{⟨p⟩}^2)}^{1∕2}$ versus logarithm of system size $L$. The disorder is $D=10$, 12, 15, and 20, respectively, and the data for each different level of disorder are shifted vertically in order to separate them. The slopes are in that order for $D=10$: 1.19, $D=12$: 1.22, $D=15$: 1.20, and $D=20$: 1.20. Taking the uncertainty in the determination of each individual slope into account, we estimate $1∕\nu =1.20\pm 0.06$, giving $\nu =0.83\pm 0.04$.

Figure 3

$⟨p_c⟩$ plotted against $L^{-1∕\nu }$ with $\nu =0.83$. $D=10(\times )$, $D=12\left(▵\right)$, $D=15\left(엯\right)$, and $D=20(+)$. As $L\to \infty$ the straight lines extrapolate to the thresholds $p_c=0.578$, $p_c=0.590$, $p_c=0.604$, and $p_c=0.619$, respectively.

Figure 4

We plot $W_c$ as a function of $⟨p⟩$. The data are the same as those used in Figs. 2, 3.

Figure 5

$p$ plotted against $1∕D$ and extrapolated to infinite disorder giving $p_c\left(\infty \right)=0.66$. Extrapolating the straight line, towards increasing $1∕D$ values, we find that $p_c$ becomes negative for $1∕D>0.75$.