Self-Repelling Bi-Exploration Process

TL;DR

This paper studies a self-repelling two-leg spider walk model with stochastic control parameters, revealing a crossover between regimes where the walker either maintains a fixed distance or collapses, affecting growth and fractal properties.

Contribution

It introduces a novel two-parameter model of self-repelling walks and characterizes the dynamic crossover between distinct behavioral regimes over time.

Findings

In the small $eta_d$ regime, the distance remains non-zero over time.

In the large $eta_d$ regime, the distance decays as a power-law with log time.

Growth process analysis shows different scaling behaviors in the two regimes.

Abstract

Self-repelling two-leg (biped) spider walk is considered where the local stochastic movements are governed by two independent control parameters $ \beta_d$ and $ \beta_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 $\beta_d$ and $\beta_h$. Our statistical analysis reveals that the system undergoes a crossover between two (small and large $\beta_d$) regimes identified in large times ($t$). In the small $\beta_d$ regime, the random walkers (identified by the position of the legs of the spider) remain on average in a fixed non-zero distance in the large time limit, whereas in the second regime (large $\beta_d$s), 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 power-law with respect to $t$ in the first regime, and $\log t$ in the second regime. The fractal dimension of the random walker traces and the winding angle are shown to consistently undergo a similar crossover.