Maxima of two random walks: universal statistics of lead changes

Ben‐Naim, E.; Krapivsky, P. L.; Randon‐Furling, Julien

doi:10.1088/1751-8113/49/20/205003

Cited by 6 publications

(5 citation statements)

References 45 publications

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“…Various reaction diffusion systems have been studied with different values of k and l in the past for different dynamical processes like ballistic annihilation [18][19][20], Levy walks [21,22] and of course simple diffusion. However, what happens if the dynamical process is intrinsically stochastic and diffusive is an important question which has not been studied much.…”

Section: Introductionmentioning

confidence: 99%

Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

Roy¹,

Ray²,

Sen³

2020

J. Phys. A: Math. Theor.

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Dynamical features of tagged particles are studied in a one dimensional A + A → kA system for k = 0 and 1, where the particles A have a bias ǫ (0 ≤ ǫ ≤ 0.5) to hop one step in the direction of their nearest neighboring particle. ǫ = 0 represents purely diffusive motion and ǫ = 0.5 represents purely deterministic motion of the particles. We show that for any ǫ, there is a time scale t * which demarcates the dynamics of the particles. Below t * , the dynamics are governed by the annihilation of the particles, and the particle motions are highly correlated, while for t ≫ t * , the particles move as independent biased walkers. t * diverges as (ǫc − ǫ) −γ , where γ = 1 and ǫc = 0.5. ǫc is a critical point of the dynamics. At ǫc, the probability S(t), that a walker changes direction of its path at time t, decays as S(t) ∼ t −1 and the distribution D(τ ) of the time interval τ between consecutive changes in the direction of a typical walker decays with a power law as D(τ ) ∼ τ −2 .

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Section: Introductionmentioning

confidence: 99%

Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

Roy¹,

Ray²,

Sen³

2020

J. Phys. A: Math. Theor.

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“…The logarithmic growth of the number of ties in one dimension resembles the growth law (1) corresponding to ties between maxima. The probability to observe n ties between maxima of two random walks during the time interval (0, t) was found to be Poissonian ∼ t −1/4 (ln t) n [12]. We anticipate a similar functional form holds for ties between the ranges of two random walks,…”

Section: The Number Of Tiesmentioning

confidence: 76%

“…To obtain the asymptotic behavior of the average number of ties, we use the general formula [12] dA dt = 2…”

Section: The Number Of Tiesmentioning

confidence: 99%

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Statistical properties of sites visited by independent random walks

Ben‐Naim¹,

Krapivsky²

2022

J. Stat. Mech.

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The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on hyper-cubic lattices and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0, t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d = 1, 2, 3, 4. We also study the time evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as a 1 ln t with a 1 = 0.970 508 in one dimension and as (ln t)2 in two dimensions.

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“…Note that this area is still an active field of research with applications in different fields. See for example [16] on an application of Stein's method on this parameters in which bounds for the convergence rate in the Kolmogorov and the Wasserstein metric are derived, [11] where the maxima of two random walks are analyzed, and [15] for applications to machine learning.…”

Section: Applications To Lattice Path Countingmentioning

confidence: 99%

A half-normal distribution scheme for generating functions

Wallner

2020

European Journal of Combinatorics

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We present an extension of a theorem by Michael Drmota and Michèle Soria [Images and Preimages in Random Mappings, 1997] which can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytic and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of lattice paths.

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Maxima of two random walks: universal statistics of lead changes

Cited by 6 publications

References 45 publications

Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

Tagged particle dynamics in one dimensional ${A+ A \to kA}$ models with the particles biased to diffuse towards their nearest neighbour

Statistical properties of sites visited by independent random walks

A half-normal distribution scheme for generating functions

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