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

TL;DR

This paper introduces a continuous-time model of random walks with memory and preferential site revisitation, revealing ultraslow diffusion and positive recurrence time correlations, supported by analytical and numerical analysis.

Contribution

It extends a discrete-time model to continuous time, providing analytical insights into ultraslow diffusion and recurrence time correlations in non-Markovian random walks.

Findings

Mean square displacement grows logarithmically over time.

Positive correlations exist between consecutive recurrence times.

Numerical results confirm ultraslow dynamics and correlation signatures.

Abstract

Random walks with memory typically involve rules where a preference for either revisiting or avoiding those sites visited in the past are introduced somehow. Such effects have a direct consequence on the statistics of first-passage and subsequent recurrence times through a site; typically, a preference for revisiting sites is expected to result in a positive correlation between consecutive recurrence times. Here we derive a continuous-time generalization of the random walk model with preferential relocation to visited sites proposed in [Phys. Rev. Lett. 112, 240601] to explore this effect, together with the main transport properties induced by the long-range memory. Despite the highly non-Markovian character of the process, our analytical treatment allows us to (i) observe the existence of an asymptotic logarithmic (ultraslow) growth for the mean square displacement, in accordance to the results found for the original discrete-time model, and (ii) confirm the existence of positive correlations between first-passage and subsequent recurrence times. This analysis is completed with a comprehensive numerical study which reveals, among other results, that these correlations between first-passage and recurrence times also exhibit clear signatures of the ultraslow dynamics present in the process.