Sadao Isogami

We studied competing directed percolation (CDP) to confirm the validity of Matsushita and Meakin's theory also for the case that an entire pattern is self-similar and not compact, while its individual clusters are self-affine. Critical values of the bond concentration p(c) and values of the various scaling exponents of CDP in 2 approximately 6 dimensions were estimated by simulations. The present result shows that Matsushita and Meakin's theory is valid for CDP in any dimension. It is also found that the estimated values of the scaling exponent of the cluster-size distribution tau are different from those of ordinary directed percolation.

Structural and statistical properties of directed percolation in two dimensions are studied numerically. The self-affinity of individual clusters is confirmed and their scaling exponents are obtained. An exponent characterizing the cluster-size distribution is also obtained. We confirm a scaling law to be valid for these exponents.

We study structural and statistical properties of a self-avoiding version of fractional Brownian motion (fBm) trajectories in two dimensions. The self-similarity of fBm trajectories is confirmed and their fractal dimensions D are obtained. Closed loops of the fBm trajectories (self-avoiding fBm) are generated, and the relationship between their fractal dimensions D(SA) and the Hurst exponent H characterizing fBm curves is obtained. It is found that the size distribution of the loops of fBm obeys a power law with the exponent f. The dependence of f on H is determined numerically.