We study the size distribution of power blackouts for the Norwegian and North American power grids. We find that for both systems the size distribution follows power laws with exponents −1.65 ± 0.05 and −2.0 ± 0.1, respectively. We then present a model with global redistribution of the load when a link in the system fails which reproduces the power law from the Norwegian power grid if the simulation are carried out on the Norwegian high-voltage power grid. The model is also applied to regular and irregular networks and give power laws with exponents −2.0 ± 0.05 for the regular networks and −1.5 ± 0.05 for the irregular networks. A presented mean-field theory is in good agreement with these numerical results.

We measure the roughness exponent for fracture profiles in the two-dimensional central force lattice model using different measurement methods. We find that the profiles are self-affine for a system with narrow disorders and that broader disorders introduces overhangs in the fracture surface leading to deviation from self-affinity for small length scales and to nontrivial finite size scaling.

We study the growth of fractal clusters in the dielectric breakdown model (DBM) by means of iterated conformal mappings. In particular we investigate the fractal dimension and the maximal growth site (measured by the Hoelder exponent alpha_{min} ) as a function of the growth exponent eta of the DBM model. We do not find evidence for a phase transition from fractal to nonfractal growth for a finite eta value. Simultaneously, we observe that the limit of nonfractal growth (D-->1) is consistent with alpha_{min}-->12 . Finally, using an optimization principle, we give a recipe on how to estimate the effective value of eta from temporal growth data of fractal aggregates.

The morphology of brittle fracture surfaces are self affine with roughness exponents that may be classified into a small number of universality classes. We discuss these in light of the recent proposal that the self affinity is a manifestation of the fracture process being a correlated percolation process. We also study numerically with high precision the roughness exponent in the two-dimensional fuse model with disorder both in breaking thresholds and conductances of the fuses. Our results are consistent with the predictions of the correlated percolation theory.

The roughness exponent for fracture surfaces in the fuse model has been thought to be universal for narrow threshold distributions and has been important in the numerical studies of fracture roughness. We show that the fuse model gives a disorder dependent roughness exponent for narrow disorders when the lattice is influencing the fracture growth. When the influence of the lattice disappears, the local roughness exponent approaches zeta(local)=0.65+/-0.03 for distribution with a tail toward small thresholds, but with large jumps in the profiles giving corrections to scaling on small scales. For very broad disorders the distribution of jumps becomes a Lévy distribution and the Lévy characteristics contribute to the local roughness exponent.