Non-Gaussian propagator for elephant random walks

Silva, M. A. A. da; Cressoni, J. C.; Schütz, Gunter M.; Viswanathan, G. M.; Trimper, Steffen

doi:10.1103/physreve.88.022115

Cited by 41 publications

(50 citation statements)

References 26 publications

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“…This latter form contrasts with the scaling functions of Markovian RW models exhibiting logarithmic diffusion (e.g., the Sinai model [44][45][46]) or stopped diffusion (e.g., the RW stochastically reset to the origin [31]), which have exponential tails. Likewise, the scaling function of the elephant walk model [23] in the anomalous regime is not Gaussian, although its precise form is not known [47]. Our results point out a new mechanism for the emergence of Gaussian distributions, which could be generic in stochastic processes where a recurrent memory does not prevent fluctuations from diverging with time, but make them grow slower than a power-law.…”

Section: (V)mentioning

confidence: 81%

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

2014

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Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the Central Limit Theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.PACS numbers: 05.40.Fb, 89.75.Fb, 87.23.Ge The individual displacements of living organisms exhibit rich statistical features over multiple temporal and spatial scales. Due to their seemingly erratic nature, animal movements are often interpreted as random search processes and modeled as random walks [1][2][3]. In recent years, the increasing availability of data on animal [4][5][6][7] as well as human [8][9][10][11] mobility have motivated numerous models inspired from the simple random walk (RW). Let us mention, in particular, multiple scales RWs, such as Lévy walks [12,13] or intermittent RWs [4,[14][15][16], which are walks with short local movements mixed with less frequent but longer commuting displacements.Markovian RWs are the basic paradigm for modeling animal and human mobility and they provide useful insights at short temporal scales. However, empirical studies conducted over long periods of times reveal pronounced non-Markovian effects [11,17,18]. As for humans, mounting evidence shows that many animals have sophisticated cognitive abilities and use memory to move to familiar places that are not in their immediate perception range [19,20]. The use of long-term memory should strongly impact movement and it is probably at the origin of many observations which are incompatible with RWs predictions, such as, very slow diffusion, heterogeneous space use, the tendency to revisit often particular places at the expense of others, or the emergence of routines [10,11,17,18,21,22]. Non-Markovian random walks where movement steps depend on the whole path of the walker [23-25] offer a promising modeling framework in this context. But the relative lack of available analytical results in this area limits the understanding of the effects of memory on mobility patterns.Self-attracting or reinforced RWs are an important class of non-Markovian dynamics [26]. In these processes, typically, a walker on a lattice moves to a nearestneighbor site with a probability that depends on the number of times this site has been visited in the past [27][28][29]. These walks must be in principle described by a hierarchy of multiple-time distribution functions, or can be studied within field theory approaches [30]. In a slightly different conte...

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Section: (V)mentioning

confidence: 81%

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

2014

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“…The ERW has been received considerable attention in the literature in the last years, and results on it range from exact moments [4,9,11] to large deviation results [7] up to connection with bond percolation on random recursive trees [8] and Pólya urn-types [1]. The extension of the ERW of [6] exhibits also subdiffusion.…”

Section: Introduction and Resultsmentioning

confidence: 99%

A strong invariance principle for the elephant random walk

Coletti

Gava²,

Schütz

2017

J. Stat. Mech.

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We consider a non-Markovian discrete-time random walk on Z with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable scaling and in the diffusive regime as well as at the critical value pc = 3/4 where the model is marginally superdiffusive, the ERW is almost surely well approximated by a Brownian motion. As a by-product of our result we get the law of iterated logarithm and the central limit theorem for the ERW.

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“…[57] for a discussion on the socalled elephant walk). The knowledge of the scaling function here has allowed us to establish useful connections between a reinforced walk and well-known Markovian models of anomalous diffusion: namely the CTRW (1 < β < 2) and the SBM (β < 1).…”

Section: Discussionmentioning

confidence: 98%

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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Non-Gaussian propagator for elephant random walks

Cited by 41 publications

References 26 publications

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

Random Walks with Preferential Relocations to Places Visited in the Past and their Application to Biology

A strong invariance principle for the elephant random walk

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

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