Laplacian fractal growth in media with quenched disorder

We study the combined effect of a Laplacian field and quenched disorder in the generation of fractal structures in the quenched dielectric breakdown model. The growth dynamics is shown to evolve from the avalanches of invasion percolation (IF) to the smooth growth of Laplacian fractals, i.e., diffusion limited aggregation and the dielectric breakdown model (DBM). The fractal dimension is strongly reduced with respect to both the DBM and IF, due to the combined effect of memory and field screening. This implies a specific relation between the fractal dimension of the breakdown structures and the microscopic properties of disordered materials.