On the fluctuations of sums of random variables

Andersen, E

doi:10.7146/math.scand.a-10385

Cited by 270 publications

(212 citation statements)

References 8 publications

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“…A link between the discretized stationary interface and a discrete random walk is thus made by means of the Sparre-Andersen theorem. 32 This theorem describes the persistence probability P 0 (n) of a random walker to stay positive (or negative) up to a step n starting in 0. We discuss as well the influence of the boundary conditions on the correlator of consecutive jumps in the interface.…”

Section: 16mentioning

confidence: 99%

Distribution of zeros in the rough geometry of fluctuating interfaces

2016

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We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce non-trivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium non-stationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the Von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.

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Section: 16mentioning

confidence: 99%

Distribution of zeros in the rough geometry of fluctuating interfaces

2016

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“…there is a useful relation due to Sparre Andersen [25]: 11) which has been used recently in Ref [26].…”

Section: Appendix: Surviving Probability Of Lévy Flightsmentioning

confidence: 99%

Random quantum magnets with broad disorder distribution

Karevski¹,

Lin²,

Rieger³

et al. 2001

Eur. Phys. J. B

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Abstract. We study the critical behavior of Ising quantum magnets with broadly distributed random couplings (J), such that P (ln J) ∼ | ln J| −1−α , α > 1, for large | ln J| (Lévy flight statistics). For sufficiently broad distributions, α < αc, the critical behavior is controlled by a line of fixed points, where the critical exponents vary with the Lévy index, α. In one dimension, with αc = 2, we obtaind several exact results through a mapping to surviving Riemann walks. In two dimensions the varying critical exponents have been calculated by a numerical implementation of the Ma-Dasgupta-Hu renormalization group method leading to αc ≈ 4.5. Thus in the region 2 < α < αc, where the central limit theorem holds for | ln J| the broadness of the distribution is relevant for the 2d quantum Ising model. PACS. 75.50.Lk Spin glasses and other random magnets -05.30.Ch Quantum ensemble theory -75.10.Nr Spin-glass and other random models -75.40.Gb Dynamic properties (dynamic susceptibility, spin waves, spin diffusion, dynamic scaling, etc.)

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“…First, let q(l) denote the probability that a walk, starting initially at x, stays above (or below) its starting position x up to step l. Clearly q(l) does not depend on the starting position x. A nontrivial theorem due to Sparre Andersen [21] states that q(l) = 2l l 2 −2l is universal for all l, i.e., independent of φ(η) as long as φ(η) is symmetric and continuous. Its generating function is simplỹ…”

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confidence: 99%

Universal Record Statistics of Random Walks and Lévy Flights

2008

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We compute exactly the statistics of the number of records in a discrete-time random walk model on a line where the walker stays at a given position with a nonzero probability 0 ≤ p ≤ 1, while with the complementary probability 1 − p, it jumps to a new position with a jump length drawn from a continuous and symmetric distribution f 0 (η). We show that, for arbitrary p, the statistics of records up to step N is completely universal, i.e., independent of f 0 (η) for any N .We also compute the connected two-time correlation function C p (m 1 , m 2 ) of the record-breaking events at times m 1 and m 2 and show it is also universal for all p. Moreover, we demonstrate that C p (m 1 , m 2 ) < C 0 (m 1 , m 2 ) for all p, indicating that a nonzero p induces additional anti-correlations between record events. We further show that these anti-correlations lead to a drastic reduction in the fluctuations of the record numbers with increasing p. This is manifest in the Fano factor, i.e. the ratio of the variance and the mean of the record number, which we compute explicitly. We also show that an interesting scaling limit emerges when p → 1, N → ∞ with the product t = (1 − p) N fixed. We compute exactly the associated universal scaling functions for the mean, variance and the Fano factor of the number of records in this scaling limit.

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On the fluctuations of sums of random variables

Cited by 270 publications

References 8 publications

Distribution of zeros in the rough geometry of fluctuating interfaces

Distribution of zeros in the rough geometry of fluctuating interfaces

Random quantum magnets with broad disorder distribution

Universal Record Statistics of Random Walks and Lévy Flights

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