Percolation in sign-symmetric random fields: Topological aspects and numerical modeling

The topology of percolation in random scalar fields psi(x) with sign symmet ry [i.e., that the statistical properties of the functions psi(x) and- psi( x) are identical] is analyzed. Based on methods of general topology, we sho w that the zero set psi(x) = 0 of the n-dimensional (n greater than or equa l to 2) sign-symmetric random field psi(x) contains a (connected) percolati ng subset under the condition /del psi(x)/not equal 0 everywhere except in domains of negligible measure. The fractal geometry of percolation is analy zed in more detail in the particular case of the two-dimensional (n=2) fiel ds psi(x). The improved Alexander-Orbach conjecture [Phys. Rev. E 56, 2437 (1997)] is applied analytically to obtain estimates of the main fractal cha racteristics of the percolating fractal sets generated by the horizontal "c uts," psi(x)= h, of the field psi(x). These characteristics are the Hausdor ff fractal dimension of the set, D, and the index of connectivity, theta. W e advocate an unconventional approach to studying the geometric properties of fractals, which involves methods of homotopic topology. It is shown that the index of connectivity, theta, of a fractal set is the topological inva riant of this set, i.e., it remains unchanged under the homeomorphic deform ations of the fractal. This issue is explicitly used in our study to find t he Hausdorff fractal dimension of the single isolevels of the field psi(x), as well as the related geometric quantities. The results obtained are anal yzed numerically in the particular case when the random field psi(x) is giv en by a fractional Brownian surface whose topological properties recover we ll the main assumptions of our consideration.