Anomalous roughness of fracture surfaces in 2D fuse models

We study anomalous scaling and multiscaling of two-dimensional crack profiles in the random fuse model using both periodic and open boundary conditions. Our large scale and extensively sampled numerical results reveal the importance of crack branching and coalescence of microcracks, which induce jumps in the solid-on-solid crack profiles. Removal of overhangs (jumps) in the crack profiles eliminates the multiscaling observed in earlier studies and reduces anomalous scaling. We find that the probability density distribution $p(Δh(\ell))$ of the height differences $Δh(\ell) = [h(x+\ell) - h(x)]$ of the crack profile obtained after removing the jumps in the profiles has the scaling form $p(Δh(\ell)) = ^-1/2 ~f(\frac{Δh(\ell)}{^1/2})$, and follows a Gaussian distribution even for small bin sizes $\ell$. The anomalous scaling can be summarized with the scaling relation $[\frac{^1/2}{^1/2}]^{1/ζ_loc} + \frac{(\ell-L/2)^2}{(L/2)^2} = 1$, where $^1/2 \sim L^ζ$.