Self-attracting walk with nu &lt;1/2

Sapozhnikov, Victor B.

doi:10.1088/0305-4470/27/6/001

Cited by 39 publications

(52 citation statements)

References 5 publications

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“…The present model exhibits distinctive features, such as a new scaling function defined through Eq. (18), and anticorrelated Lévy steps with exponent −3 at the subdiffusive transition. The visitation and first-passage statistics are also likely to be peculiar in this model, and their study certainly deserves future work.…”

Section: Discussionmentioning

confidence: 99%

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

Boyer

Romo-Cruz

2014

Phys. Rev. E

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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) but also the propagator can be obtained exactly in the asymptotic limit. 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 anticorrelated 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.

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Section: Discussionmentioning

confidence: 99%

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

Boyer

Romo-Cruz

2014

Phys. Rev. E

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“…Random walk models in similar vein to ours had been studied in relation to self-interacting random walks [8][9][10][11][12]. For example, Duxbury and de Queiroz [10] assigned to each (simple) random walk trajectory the "energy" in the form of exp (−g i n i α ), where n i is the visitation numbers of the site i as in our model.…”

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confidence: 92%

Modeling the mobility with memory

Choi

Sohn

Goh

et al. 2012

EPL

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PACS 89.75.Da -Systems obeying scaling laws PACS 89.65.-s -Social and economic systems Abstract. -We study a random walk model in which the jumping probability to a site is dependent on the number of previous visits to the site, as a model of the mobility with memory. To this end we introduce two parameters called the memory parameter α and the impulse parameter p. From extensive numerical simulations, we found that various limited mobility patterns such as subdiffusion, trapping, and logarithmic diffusion could be observed. By the memory, a long-ranged directional anti-correlation kinetically-induces anomalous sub-diffusive and trapping behaviors, and transition between them. With random jumps by the impulse parameter, a trapped walker can escape from the trap very slowly, resulting in an ultraslow logarithmic diffusive behavior. Our results suggest that the memory of walker's has-beens can be one mechanism explaining many of empirical characteristics of the mobility of animated objects.

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“…Self-attracting random walks are path-dependent processes where a walker tends to return to previously visited sites [36,37]. Numerical simulations and scaling arguments clearly show that selfattracting walks can exhibit subdiffusion [38][39][40]. These mathematically challenging processes cannot be readily analyzed with better known frameworks for subdiffusive phenomena, such as fractional Fokker-Planck equations [8] or scaled Brownian motions [41,42].…”

Section: Introductionmentioning

confidence: 99%

Slow Lévy flights

Boyer

Pineda

2016

Phys. Rev. E

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Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-thanBrownian dynamics. Here we show that Lévy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very slow diffusion, logarithmic in time. These processes are path-dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the Central Limit Theorem is modified in this context, keeping the usual distinction between analytic and non-analytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

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Self-attracting walk with nu <1/2

Cited by 39 publications

References 5 publications

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

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

Modeling the mobility with memory

Slow Lévy flights

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