Random geometric series

Ben‐Naim, E.; Krapivsky, P. L.

doi:10.1088/0305-4470/37/23/001

Cited by 7 publications

(9 citation statements)

References 16 publications

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“…for 1 ≤ j ≤ d − 1. We can verify that the average number of points is consistent with the exact behavior N −1/d dy Q j (y j ) ∼ (ln N ) d−1 as in (19). The crossover moment vanishes and the moments decay logarithmically,…”

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confidence: 76%

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Statistics of partial minima

Ben‐Naim¹,

Hastings²,

Izraelevitz³

2007

J. Phys. A: Math. Theor.

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Motivated by multi-objective optimization, we study extrema of a set of N points independently distributed inside the d-dimensional hypercube. A point in this set is k-dominated by another point when at least k of its coordinates are larger, and is a k-minimum if it is not k-dominated by any other point. We obtain statistical properties of these partial minima using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ∼ N −(d−k)/k , when 1 ≤ k < d. Interestingly, there are k − 1 distinct scaling laws characterizing the largest coordinates as the distribution P (yj) of the jth largest coordinate, yj, decays algebraically, P (yj) ∼ (yj ) −α j −1 , with αj = j d−k k−j for 1 ≤ j ≤ k − 1. The average number of partial minima grows logarithmically, A ≃ 1 (d−1)! (ln N ) d−1 , when k = d. The full distribution of the number of minima is obtained in closed form in two-dimensions.

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confidence: 76%

“…Another interesting issue is the crossover from the algebraic decay (2) to the logarithmic growth (19). The average number of partial minima decreases monotonically with N when k is small, but is a non-monotonic function of N when k is large.…”

mentioning

confidence: 99%

Statistics of partial minima

Ben‐Naim¹,

Hastings²,

Izraelevitz³

2007

J. Phys. A: Math. Theor.

Self Cite

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“…The average and variance of the number of fragments follow the same law as for the problem of random geometric series [37]. Indeed, if one takes for example the time series x t = 2x p , with 0 ≤ p ≤ t − 1 chosen randomly and initial condition x 0 = 1, considering instead the variable n t = log(x t )/ log(2) leads to the recurrence n t = n p + 1.…”

Section: Fragment Number Distributionmentioning

confidence: 89%

“…The distribution of the number of fragments is therefore related to a record process. Another class of stochastic models in close relation with Stirling numbers concerns random geometric series [36,37], where a growing sequence of terms {x n } n=0,..,t at time t is constructed from a random process, for example x t = 2x p , with x p , 0 ≤ p ≤ t − 1, one of the previous term chosen with a given probability and multiplied by a factor 2. In this case the probability distribution of the logarithmic sequence depends explicitly on Stirling numbers.…”

Section: Fragment Number Distributionmentioning

confidence: 99%

Modified stochastic fragmentation of an interval as an ageing process

Fortin¹

2018

J. Stat. Mech.

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We study a stochastic model based on a modified fragmentation of a finite interval. The mechanism consists in cutting the interval at a random location and substituting a unique fragment on the right of the cut to regenerate and preserve the interval length. This leads to a set of segments of random sizes, with the accumulation of small fragments near the origin. This model is an example of record dynamics, with the presence of "quakes" and slow dynamics. The fragment size distribution is a universal inverse power law with logarithmic corrections. The exact distribution for the fragment number as function of time is simply related to the unsigned Stirling numbers of the first kind. Two-time correlation functions are defined and computed exactly. They satisfy scaling relations and exhibit aging phenomena. In particular the probability that the same number of fragments is found at two different times t > s is asymptotically equal to [4π log(s)] −1/2 when s ≫ 1 and the ratio t/s fixed, in agreement with the numerical simulations. The same process with a reset impedes the aging phenomena beyond a typical time scale defined by the reset parameter.

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“…In Ben- Naim & Krapivsky (2004), a further range of history-dependent sequences was examined, based on uniform random selection from the past. Moving away from the assumption of uniformity, Krasikov et al (2004) considered more general sequences of the form X nC1 Z X n C bX H ðnÞ ; ð1:5Þ…”

Section: Introductionmentioning

confidence: 99%

History-dependent random processes

Clifford

Stirzaker

2008

Proc. R. Soc. A.

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Ulam has defined a history-dependent random sequence by the recursion X nC1 ZX n C X U(n) , where (U(n); nR1) is a sequence of independent random variables with U(n) uniformly distributed on {1, ., n} and X 1 Z1. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam's sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discretetime formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease.

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Random geometric series

Cited by 7 publications

References 16 publications

Statistics of partial minima

Statistics of partial minima

Modified stochastic fragmentation of an interval as an ageing process

History-dependent random processes

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