Narrow log-periodic modulations in non-Markovian random walks

Area of Science:

• Complex Systems
• Statistical Mechanics
• Dynamical Processes

Background:

• Log-periodic oscillations are typically associated with negative feedback and long-range memory in random walk models.
• Previous research identified negative feedback as a crucial component for generating these oscillations.

Purpose of the Study:

• To investigate the conditions under which log-periodicity emerges in random walk dynamics.
• To determine if positive feedback can also lead to log-periodic behavior.
• To explore the role of memory correlations in the emergence of these oscillations.

Main Methods:

• Utilizing discrete-time random walk models with strong memory correlations.
• Employing memory profiles based on binomial and delta distributions.
• Analyzing the system's behavior in the large time limit.

Main Results:

• Demonstrated that small-amplitude log-periodic oscillations can emerge even with positive feedback.
• Showcased anomalous superdiffusive behavior in the large time limit.
• Identified log-periodic modulations arising from specific parameter choices in the models.

Conclusions:

• Positive feedback, contrary to previous assumptions, can drive log-periodic oscillations in random walks.
• These oscillations may be subtle and easily misidentified as finite-size effects.
• Memory correlations play a significant role in the emergence of anomalous diffusion and log-periodicity.