Asymmetric Lévy flights in the presence of absorbing boundaries

Mulatier, Clélia de; Rosso, Alberto; Schehr, Grégory

doi:10.1088/1742-5468/2013/10/p10006

Cited by 16 publications

(18 citation statements)

References 25 publications

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“…The agreement between the distribution ofR M and the distribution of the total displacements for large R is clearly visible and both distributions converge to the asymptotic results in Fig. 4 shows indeed that the biggest jumpR M and the final position R are correlated: each dot representR M and R for a single walker at T = 2 16 . For large R we observe thatR M ≈ R, while for short distances large fluctuations are present due to multiple processes.…”

Section: Lévy Walks: the Jump Rate And The Big Jumpsupporting

confidence: 72%

“…From Eqs. (14)(15)(16)(17)(18)(19) we also get that the probability that a step has length L is q(L) ∼ L −ν−1 .…”

Section: Anomalous Diffusion For Cold Atoms In Optical Latticesmentioning

confidence: 70%

“…(1) when X is large. This means that the tail of the distribution of the largest summand is the same as the tail of the sum and in this sense the sum is dominated by a single macroscopic jump ( [15][16][17][18]). An example in the IID domain are Lévy flights in dimension one, which deal with a sum of displacements drawn from a power law probability density function λ(x i ) ∼ x −(1+α) i , and in this case one finds that Eq.…”

Section: Introductionmentioning

confidence: 99%

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Single-big-jump principle in physical modeling

2019

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The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell. PACS numbers:

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Section: Lévy Walks: the Jump Rate And The Big Jumpsupporting

confidence: 72%

“…From Eqs. (14)(15)(16)(17)(18)(19) we also get that the probability that a step has length L is q(L) ∼ L −ν−1 .…”

Section: Anomalous Diffusion For Cold Atoms In Optical Latticesmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

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Single-big-jump principle in physical modeling

2019

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“…for some c > 0 and any fixed x. Substituting (61) for f (g + x) on the right-hand side of (17) and using (18), one gets…”

Section: µ = 1 and Summarymentioning

confidence: 99%

“…The reason for such a choice is twofold: (i) random numbers drawn from (99) are easy and fast to generate numerically (see for instance Ref. [17]) and (ii) f (η) is a power law for all |η| ≥ 1 (instead of |η| 1 ) which allows us to access the asymptotic regime characterizing Lévy flights rather quickly in numerical simulations.…”

Section: Numerical Simulationsmentioning

confidence: 99%

On the gap and time interval between the first two maxima of long continuous time random walks

Mounaix¹,

Schehr²,

Majumdar³

2016

J. Stat. Mech.

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We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval T where at each time step the walker waits a random time τ , before performing a jump drawn from a symmetric continuous probability distribution function (PDF) f (η), of Lévy index 0 < µ ≤ 2.Our study includes the case where the waiting time PDF Ψ(τ ) has a power law tail, Ψ(τ ) ∝ τ −1−γ , with 0 < γ < 1, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if γ < µ/2 (and super-diffusive if γ > µ/2). We investigate the joint PDF of the gap g between the first two highest positions of the CTRW and the time t separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution p(g, t) in the limit of long CTRW, T → ∞. Our exact analytical results show a very rich behavior of this joint PDF in the (γ, µ) plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).

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