Self-similarity and a phase-transition-like behavior of a random growing structure governed by a nonequilibrium parameter

Fractal properties of a random pattern formation produced by computer simulation have been analyzed. The controlling parameter was a tip priority factor R by which a growing tip grows further compared to any site on a branch to produce a new side branch. The pattern was found to show an approximate self-similarity and associated with two fractal dimensions: One is the inner dimension which measures the fine structure and the other is the outer dimension which measures the framework of the pattern. With increasing tip priority factor, the inner dimension decreases and shows a phase-transition-like behavior at R=35. It shows damped oscillations approaching a constant value as the volume is increased. The outer dimension roughly remains 2. A possible mechanism for these behaviors is discussed.