Stationary self-organized fractal structures in an open, dissipative electrical system

We study the boundary effects in invasion percolation (IP) with and without trapping. We find that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show a fractal dimension, for the region of the percolating cluster near the boundary, remarkably different from the bulk one. In fact, on the surface we find a value ofDsur D 1:65 0:02 (for IP with trapping Dsur tr D 1:59 0:03), compared with the bulk value of Dbul D 1:880:02 (Dbul tr D 1:850:02). We find a logarithmic crossover from surface to bulk fractal properties, as one would expect from the finite-size theory of critical systems. The distribution of the quenched variables on the growing interface near the boundary self-organizes into an asymptotic shape characterized by a discontinuity at a value xc D 0:5, which coincides with the bulk critical threshold. The exponent sur of the boundary avalanche distribution for IP without trapping is sur D 1:56 0:05; this value is very near to the bulk one. Then we conclude that only the geometrical properties (fractal dimension) of the model are affected by the presence of a boundary, while other statistical and dynamical properties are unchanged. Furthermore, we are able to present a theoretical computation of the relevant critical exponents near the boundary. This analysis combines two recently introduced theoretical tools, the fixed scale transformation and the run time statistics, which are particularly suited for the study of irreversible self-organized growth models with quenched disorder. Our theoretical results are in rather good agreement with numerical data