Slow Lévy flights

Boyer, Denis; Pineda, Inti

doi:10.1103/physreve.93.022103

Cited by 24 publications

(26 citation statements)

References 58 publications

(86 reference statements)

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“…where, we recall that a and b are given in Eqs. (15) and (16). It is evident from this exact solution in Eq.…”

Section: The Modelmentioning

confidence: 58%

“…This model turns out to be exactly solvable [14] and the particle position distribution converges to a Gaussian at long times with a variance growing logarithmically as σ 2 ∼ ln t. The model was generalised in [15] to relocation to a site visited in the past, selected according to a time-weighted distribution with a specific form. This has been further generalised to memory kernels that lead to an asymptotic Lévy distribution with index 0 < µ ≤ 2 for the particle position with a length scale growing as (ln t) 1/µ [16]. Another simple model with memory that has attracted recent interest is diffusion with stochastic resetting [17,18].…”

Section: Introductionmentioning

confidence: 99%

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Long time scaling behaviour for diffusion with resetting and memory

Boyer¹,

Evans²,

Majumdar³

2017

J. Stat. Mech.

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We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive (σ 2 ∼ t) all the way to anomalous ultraslow growth σ 2 ∼ ln ln t.

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“…where, we recall that a and b are given in Eqs. (15) and (16). It is evident from this exact solution in Eq.…”

Section: The Modelmentioning

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Long time scaling behaviour for diffusion with resetting and memory

Boyer¹,

Evans²,

Majumdar³

2017

J. Stat. Mech.

Self Cite

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“…The collective model presented here is an extension of another one for a single forager with reinforcement learning, exposed under slightly different forms in [13,[54][55][56][57]. To summarize, the motion of the individuals is assumed to be driven by the combination of two basic movement modes: standard random walk displacements and preferential returns to places visited in the past.…”

Section: Modelmentioning

confidence: 99%

Collective learning from individual experiences and information transfer during group foraging

Falcón-Cortés

Boyer

Ramos‐Fernández

2019

J. R. Soc. Interface.

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Living in groups brings benefits to many animals, such as protection against predators and an improved capacity for sensing and making decisions while searching for resources in uncertain environments. A body of studies has shown how collective behaviours within animal groups on the move can be useful for pooling information about the current state of the environment. The effects of interactions on collective motion have been mostly studied in models of agents with no memory. Thus, whether coordinated behaviours can emerge from individuals with memory and different foraging experiences is still poorly understood. By means of an agent-based model, we quantify how individual memory and information fluxes can contribute to improving the foraging success of a group in complex environments. In this context, we define collective learning as a coordinated change of behaviour within a group resulting from individual experiences and information transfer. We show that an initially scattered population of foragers visiting dispersed resources can gradually achieve cohesion and become selectively localized in space around the most salient resource sites. Coordination is lost when memory or information transfer among individuals is suppressed. The present modelling framework provides predictions for empirical studies of collective learning and could also find applications in swarm robotics and motivate new search algorithms based on reinforcement.

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“…The present paper continues this line of research and considers a different special case of stochastic processes with a reset mechanism. See [12][13][14][15][16] for other developments in this regard.…”

Section: Introductionmentioning

confidence: 99%

Directed random walk with random restarts: The Sisyphus random walk

Montero¹,

Villarroel

2016

Phys. Rev. E

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In this paper we consider a particular version of the random walk with restarts: random reset events which suddenly bring the system to the starting value. We analyze its relevant statistical properties, like the transition probability, and show how an equilibrium state appears. Formulas for the first-passage time, high-water marks, and other extreme statistics are also derived; we consider counting problems naturally associated with the system. Finally we indicate feasible generalizations useful for interpreting different physical effects.

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Slow Lévy flights

Cited by 24 publications

References 58 publications

Long time scaling behaviour for diffusion with resetting and memory

Long time scaling behaviour for diffusion with resetting and memory

Collective learning from individual experiences and information transfer during group foraging

Directed random walk with random restarts: The Sisyphus random walk

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