Scale Invariance in Percolation Theory and Fractals

Details Scale Invariance in Percolation Theory and Fractals Scale Invariance in Percolation Theory and Fractals

1997-02-22

arxiv.org/abs/cond-mat/9702204v1

JETP Lett., Vol. 64, No. 6, 25 Sept. 1996

Physics

Condensed Matter

Disordered Systems and Neural Networks

LaTeX file and 2 PostScripts with 6 fig.

Scientific paper

The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot transformation, leading to a fractal behavior of the percolation probability in the complex plane. The hierarchical chains of impedances, reducing to a nonlinear mapping of the impedance space onto itself, are studied. An infinite continuation of the procedure leads to a fixed point. It is shown that the number of steps required to reach a neighborhood of this point has a fractal distribution.

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