A record-driven growth process

Godrèche, Claude; Luck, J. M.

doi:10.1088/1742-5468/2008/11/p11006

Cited by 52 publications

(83 citation statements)

References 29 publications

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“…In recent years there has been a surge of interest in the statistics of record-breaking events, both from the theoretical point of view as well as in multiple applications. The occurrence of record-breaking events has been studied, for instance, in sports [1,2], in evolution models in biology [3,4], in the theory of spin glasses [5,6], and in models of growing networks [7]. Recently there has been some progress in understanding the phenomenon of global warming via studying the occurrence of record-breaking temperatures [8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Record statistics for multiple random walks

2012

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We study the statistics of the number of records R(n,N) for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ(2) of the jump distribution is finite and (II) when σ(2) is divergent as in the case of Lévy flights with index 0<μ<2. In both cases we find that the mean record number R(n,N) grows universally as ~α(N) sqrt[n] for large n, but with a very different behavior of the amplitude α(N) for N>1 in the two cases. We find that for large N, α(N) ≈ 2sqrt[lnN] independently of σ(2) in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α(N) ≈ 4/sqrt[π], independently of 0<μ<2. For finite σ(2) we argue-and this is confirmed by our numerical simulations-that the full distribution of (R(n,N)/sqrt[n]-2sqrt[lnN])sqrt[lnN] converges to a Gumbel law as n → ∞ and N → ∞. In case II, our numerical simulations indicate that the distribution of R(n,N)/sqrt[n] converges, for n → ∞ and N → ∞, to a universal nontrivial distribution independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poor's 500 index.

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Section: Introductionmentioning

confidence: 99%

Record statistics for multiple random walks

2012

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“…The existence of connections between the two fields is well known [53,54] and they recently showed up in various problems of statistical physics [47,52]. One of the main manifestations of this connection is that the number of records R N for N i.i.d.…”

Section: Connection With Random Permutationsmentioning

confidence: 99%

“…Note that this formula (52) can also be directly obtained by noticing that the event that m < M k,n < m + dm is the same as the following event where X i ≥ m + dm for (k − 1) of the X i 's, m < X i < m + dm for exactly one of the X i 's and X i ≤ m for the remaining N − k X i 's. In particular for k = 1 one obtains from (52) the pdf of X max as…”

Section: Finite Samplementioning

confidence: 99%

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Exact Record and Order Statistics of Random Walks via First-Passage Ideas

Schehr

Majumdar

2014

First-Passage Phenomena and Their Applications

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While records and order statistics of independent and identically distributed (i.i.d.) random variables X1, · · · , XN are fully understood, much less is known for strongly correlated random variables, which is often the situation encountered in statistical physics. Recently, it was shown, in a series of works, that one-dimensional random walk (RW) is an interesting laboratory where the influence of strong correlations on records and order statistics can be studied in detail. We review here recent exact results which have been obtained for these questions about RW, using techniques borrowed from the study of first-passage problems. We also present a brief review of the well known (and not so well known) results for records and order statistics of i.i.d. variables.

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“…The recursion (2.25) has a non-polynomial solution, independent of n, 28) describing the stationary degree distribution on an infinitely large network:…”

Section: Degree Statistics Of the Whole Networkmentioning

confidence: 99%

Finite-Time Fluctuations in the Degree Statistics of Growing Networks

2009

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This paper presents a comprehensive analysis of the degree statistics in models for growing networks where new nodes enter one at a time and attach to one earlier node according to a stochastic rule. The models with uniform attachment, linear attachment (the Barabási-Albert model), and generalized preferential attachment with initial attractiveness are successively considered. The main emphasis is on finite-size (i.e., finite-time) effects, which are shown to exhibit different behaviors in three regimes of the size-degree plane: stationary, finite-size scaling, large deviations.PACS 64.60.aq, 89.75.Hc,

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A record-driven growth process

Cited by 52 publications

References 29 publications

Record statistics for multiple random walks

Record statistics for multiple random walks

Exact Record and Order Statistics of Random Walks via First-Passage Ideas

Finite-Time Fluctuations in the Degree Statistics of Growing Networks

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