Resistor-network approach to growth probability for dielectric-breakdown models at a surface

For the dielectric-breakdown model a general method is presented for calculation of the growth probability on the growing-perimeter sites of the aggregate at a surface. The growth probability is given by solving the electrostatic problem for a superconducting aggregate in the semi-infinite normal-resistor network. The electric field at the perimeter site is obtained by solving the resistor-network problem with the lattice bond-bond Green’s function. The fractal dimension D is found from the scaling assumption for the cluster-top occupancy probability.