First gap statistics of long random walks with bounded jumps

Mounaix, Philippe; Schehr, Grégory

doi:10.1088/1751-8121/aa65f2

Cited by 3 publications

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“…( 26)]. Of particular interest in this context are the gaps between two successive maxima and for which a lot of results have been obtained during the last years [104,[138][139][140][141][142][143][144][145].…”

Section: One-dimensional Brownian Motion and Random Walksmentioning

confidence: 99%

Extreme value statistics of correlated random variables: A pedagogical review

2020

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Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of 'uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fréchet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.

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Section: One-dimensional Brownian Motion and Random Walksmentioning

confidence: 99%

Extreme value statistics of correlated random variables: A pedagogical review

2020

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Record statistics of a strongly correlated time series: random walks and Lévy flights

Godrèche

Majumdar

Schehr

2017

J. Phys. A: Math. Theor.

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Abstract. We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful "laboratory" to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statistics of the number of records, as well as of the ages of the records, i.e., the lapses of time between two successive record breaking events. Then we review the results that were obtained for a wide variety of random walk models, including random walks with a linear drift, continuous time random walks, constrained random walks (like the random walk bridge) and the case of multiple independent random walkers. Finally, we discuss further observables related to records, like the record increments, as well as some questions raised by physical applications of record statistics, like the effects of measurement error and noise.arXiv:1702.00586v1 [cond-mat.stat-mech]

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Survival probability of random walks and Lévy flights on a semi-infinite line

Majumdar¹,

Mounaix²,

Schehr³

2017

J. Phys. A: Math. Theor.

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We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, f (η), characterized by a Lévy index µ ∈ (0, 2], which includes standard random walks (µ = 2) and Lévy flights (0 < µ < 2). We study the survival probability, q(x0, n), representing the probability that the RW stays non-negative up to step n, starting initially at x0 ≥ 0. Our main focus is on the x0-dependence of q(x0, n) for large n. We show that q(x0, n) displays two distinct regimes as x0 varies: (i) for x0 = O(1) ('quantum' regime), the discreteness of the jump process significantly alters the standard scaling behavior of q(x0, n) and (ii) for x0 = O(n 1/µ ) ('classical' regime) the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for µ = 2 this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in q(x0, n) occurs between the quantum and the classical regime as one increases

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First gap statistics of long random walks with bounded jumps

Cited by 3 publications

References 28 publications

Extreme value statistics of correlated random variables: A pedagogical review

Extreme value statistics of correlated random variables: A pedagogical review

Record statistics of a strongly correlated time series: random walks and Lévy flights

Survival probability of random walks and Lévy flights on a semi-infinite line

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