Self-repelling bipedal exploration process

A self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters β_{d} and β_{h}, so that the former controls the distance (d) between the legs positions, and the latter controls the statistics of self-crossing of the traversed paths. The probability measure for local movements is supposed to be the one for the "true self-avoiding walk" multiplied by a factor exponentially decaying with d. After a transient behavior for short times, a variety of behaviors have been observed for large times depending on the value of β_{d} and β_{h}. Our statistical analysis reveals that the system undergoes a crossover between two (small and large β_{d}) regimes identified in large times (t). In the small β_{d} regime, the random walkers (identified by the position of the legs of the spider) remain on average in a fixed nonzero distance in the large time limit, whereas in the second regime (large β_{d}), the absorbing force between the walkers dominates the other stochastic forces. In the latter regime, d decays in a power-law fashion with the logarithm of time. When the system is mapped to a growth process (represented by a height field which is identified by the number of visits for each point), the roughness and the average height show different behaviors in two regimes, i.e., they show a power law with respect to t in the first regime and logt in the second regime. The fractal dimension of the random walker traces and the winding angle are shown to consistently undergo a similar crossover.