Recurrence time correlations in random walks with preferential relocation to visited places

Area of Science:

• Statistical Mechanics
• Complex Systems
• Stochastic Processes

Background:

• Random walks with memory exhibit preferences for revisiting or avoiding past sites.
• These memory effects influence transport properties and first-passage/recurrence time statistics.
• A preference for revisiting sites suggests positive correlations between consecutive recurrence times.

Purpose of the Study:

• To derive a continuous-time generalization of a random walk model with preferential relocation to visited sites.
• To explore the transport properties and memory effects in this generalized model.
• To investigate the correlations between first-passage and recurrence times.

Main Methods:

• Derivation of a continuous-time random walk model with preferential relocation.
• Analytical treatment of transport properties, including mean square displacement.
• Comprehensive numerical study to analyze correlations and relaxation dynamics.

Main Results:

• An asymptotic logarithmic (ultraslow) growth for the mean square displacement was observed.
• Positive correlations between first-passage and subsequent recurrence times were confirmed.
• Numerical analysis revealed signatures of ultraslow relaxation dynamics in these correlations.

Conclusions:

• The continuous-time model successfully captures ultraslow dynamics and memory effects.
• Positive correlations between recurrence times are a key feature of this model.
• The study highlights the complex interplay between memory, transport, and relaxation in stochastic processes.