Random recursive trees and the elephant random walk

Kürsten, Rüdiger

doi:10.1103/physreve.93.032111

Cited by 54 publications

(67 citation statements)

References 36 publications

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“…It is known that globally correlated stochastic dynamics lead to anomalous diffusion processes [27][28][29][30][31][32][33][34][35]. On the other hand, we remark that the interplay between memory effects and weak ergodicity breaking was study previously such as for example in correlated continuous-time random walk models [36,37], single-file diffusion [38], and fractional Brownian-Langevin motion [39].…”

Section: Introductionmentioning

confidence: 62%

“…For alternative memory mechanisms, such as that associated to the elephant random walk model [27][28][29], the previous randomness is absent [46].…”

Section: A Asymptotic Randomnessmentioning

confidence: 99%

“…The next values are determinate by a conditional probability T (σ 1 , · · · σ t |σ t+1 ) [49] that depends on the whole previous jump trajectory: σ 1 , · · · σ t . Different memory mechanisms can be introduced through T (σ 1 , · · · σ t |σ t+1 ), such as for example in the elephant random walk model [27][28][29]. Here, we analyze an alternative urn-like dynamics [46], where…”

Section: Global Correlated Random Walk Dynamicsmentioning

confidence: 99%

“…The asymptotic property (29) implies that at large times the time-averaged moment δ κ (∆) = lim t→∞ δ κ (t, ∆) becomes a random variable. In fact, its average over realizations can be written as…”

Section: A Asymptotic Randomnessmentioning

confidence: 99%

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Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic function is calculated in an exact way, which allows us to demonstrate that the ensemble of realizations is ballistic. Asymptotically each realization is equivalent to that of a biased Markovian diffusion process with transition rates that strongly differs from one trajectory to another. Using this "inhomogeneous diffusion" feature the ergodic properties of the dynamics are analytically studied through the time-averaged moments. Even in the long time regime they remain random objects. While their average over realizations recover the corresponding ensemble averages, departure between time and ensemble averages is explicitly shown through their probability densities. For the density of the second time-averaged moment an ergodic limit and the limit of infinite lag times do not commutate. All these effects are induced by the memory effects. A generalized Einstein fluctuation-dissipation relation is also obtained for the time-averaged moments.

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Section: Introductionmentioning

confidence: 62%

“…For alternative memory mechanisms, such as that associated to the elephant random walk model [27][28][29], the previous randomness is absent [46].…”

Section: A Asymptotic Randomnessmentioning

confidence: 99%

Section: Global Correlated Random Walk Dynamicsmentioning

confidence: 99%

Section: A Asymptotic Randomnessmentioning

confidence: 99%

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Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Budini

2016

Phys. Rev. E

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“…Both claims in Theorem 3 follow from a straightforward application of Theorem 7 to our random walk. Consider K n := 2un ansn ; then we can verify that lim n→∞ K n = 0 as a consequence of (13), (14), (15) and (16). Therefore, the proof of the result in item a)…”

Section: Law Of the Iterated Logarithmmentioning

confidence: 82%

Limit theorems for a minimal random walk model

Coletti¹,

Gava²,

Lima³

2019

J. Stat. Mech.

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We study the minimal random walk introduced by Kumar, Harbola and Lindenberg [12]. It is a random process on {0, 1, . . .} with unbounded memory which exhibits subdiffusive, diffusive and superdiffusive regimes. We prove the law of large numbers for the whole parameter set. Then we prove the central limit theorem and the law of the iterated logarithm for the minimal random walk under diffusive and marginally superdiffusive behaviors. More interestingly, we establish a result for the minimal random walk when it possesses the three regimes; we show the convergence of its rescaled version to a non-normal random variable.arXiv:1908.09199v1 [math.PR]

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Broadcasting‐induced colorings of preferential attachment trees

Desmarais

Holmgren

Wagner

2023

Random Struct Algorithms

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We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability p$$ p $$ and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

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Random recursive trees and the elephant random walk

Cited by 54 publications

References 36 publications

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Inhomogeneous diffusion and ergodicity breaking induced by global memory effects

Limit theorems for a minimal random walk model

Broadcasting‐induced colorings of preferential attachment trees

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