Solvable random walk model with memory and its relations with Markovian models of anomalous diffusion

Abstract

Motivated by studies on the recurrent properties of animal and human mobility, we introduce a path-dependent random walk model with long range memory for which not only the mean square displacement (MSD) can be obtained exactly in the asymptotic limit, but also the propagator. The model consists of a random walker on a lattice, which, at a constant rate, stochastically relocates at a site occupied at some earlier time. This time in the past is chosen randomly according to a memory kernel, whose temporal decay can be varied via an exponent parameter. In the weakly non-Markovian regime, memory reduces the diffusion coefficient from the bare value. When the mean backward jump in time diverges, the diffusion coefficient vanishes and a transition to an anomalous subdiffusive regime occurs. Paradoxically, at the transition, the process is an anti-correlated Lévy flight. Although in the subdiffusive regime the model exhibits some features of the continuous time random walk with infinite mean waiting time, it belongs to another universality class. If memory is very long-ranged, a second transition takes place to a regime characterized by a logarithmic growth of the MSD with time. In this case the process is asymptotically Gaussian and effectively described as a scaled Brownian motion with a diffusion coefficient decaying as 1/t.

13 pages, 8 figures

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Cited by in corpus (26)

• Stochastic Resetting and Applications
• Elephant Random Walks and their connection to Pólya-type urns
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• Phase transitions in optimal search times: how random walkers should combine resetting and flight scales
• Ising model with stochastic resetting
• Long time scaling behaviour for diffusion with resetting and memory
• Diffusion with Partial Resetting
• The Shark Random Swim (Lévy flight with memory)
• Anderson-like localization transition of random walks with resetting
• Slow Lévy flights
• Phase transition in non-Markovian animal exploration model with preferential returns
• Anomalous diffusion in random-walks with memory-induced relocations
• A memory-induced diffusive-superdiffusive transition: ensemble and time-averaged observables
• Weak ergodicity breaking induced by global memory effects
• Inhomogeneous diffusion and ergodicity breaking induced by global memory effects
• Recurrence time correlations in random walks with preferential relocation to visited places
• Superdiffusion in self-reinforcing run-and-tumble model with rests
• Non-stationary Markovian Replication Process causing Diverse Diffusions
• Power-law relaxation of a confined diffusing particle subject to resetting with memory
• Diffusion with preferential relocation in a confining potential
• Persistence exponents of self-interacting random walks
• Ulam's History-dependent Random Adding Process
• Anomalous stochastic transport of particles with self-reinforcement and Mittag-Leffler distributed rest times
• Elephant random walks with multiple extractions and general reinforcement functions
• Ultra slow sub-logarithmic diffusion of a sluggish random walker subject to resetting with memory
• Random walkers on a deformable medium