A bernoulli excursion and its various applications

Takács, Lajos

doi:10.1017/s0001867800023739

Cited by 56 publications

(106 citation statements)

References 21 publications

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“…This limit was first proved by Takács for simple lattice random walks [33] and was recently extended to all symmetric lattice random walks with a finite variance σ [48].…”

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confidence: 84%

Conditioned random walks and interaction-driven condensation

Szavits-Nossan

Evans

Majumdar

2016

J. Phys. A: Math. Theor.

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We consider a discrete-time continuous-space random walk under the constraints that the number of returns to the origin (local time) and the total area under the walk are fixed. We first compute the joint probability of an excursion having area a and returning to the origin for the first time after time τ . We then show how condensation occurs when the total area constraint is increased: an excursion containing a finite fraction of the area emerges. Finally we show how the phenomena generalises previously studied cases of condensation induced by several constraints and how it is related to interaction-driven condensation which allows us to explain the phenomenon in the framework of large deviation theory.

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“…This limit was first proved by Takács for simple lattice random walks [33] and was recently extended to all symmetric lattice random walks with a finite variance σ [48].…”

mentioning

confidence: 84%

Conditioned random walks and interaction-driven condensation

Szavits-Nossan

Evans

Majumdar

2016

J. Phys. A: Math. Theor.

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“…The probability distribution of the area under a Brownian excursion was calculated exactly and is known as the Airy distribution function, a complicated function that involves the zeros of the Airy function but is not the Airy function itself [24,25,26,27,28]. This Airy distribution function has been a subject of intense study for the past several years as it has resurfaced in many problems in computer science [26,27], graph theory [29], and two dimensional polygon problems [30]. Recently it was shown that the same Airy distribution function also describes the distribution of maximal height in the stationary state of fluctuating interfaces [18,19].…”

Section: Exact Calculation Of the Restricted Propagator G +mentioning

confidence: 99%

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

Majumdar

Dasgupta

2006

Phys. Rev. E

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We report numerical and analytic results for the spatial survival probability for fluctuating one-dimensional interfaces with Edwards-Wilkinson or Kardar-Parisi-Zhang dynamics in the steady state. Our numerical results are obtained from analysis of steady-state profiles generated by integrating a spatially discretized form of the Edwards-Wilkinson equation to long times. We show that the survival probability exhibits scaling behavior in its dependence on the system size and the "sampling interval" used in the measurement for both "steady-state" and "finite" initial conditions. Analytic results for the scaling functions are obtained from a path-integral treatment of a formulation of the problem in terms of one-dimensional Brownian motion. A "deterministic approximation" is used to obtain closed-form expressions for survival probabilities from the formally exact analytic treatment. The resulting approximate analytic results provide a fairly good description of the numerical data.

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“…and γ = 3/2 and B(·) is a scaling function. To analytically obtain the scaling exponent γ = 3/2 note that It is interesting to note that p(χ|τ ) in the case of free diffusion (corresponding to the limit D → ∞) has been previously considered by mathematicians [26][27][28] and shown to obey the scaling relation Eq. (5), with B given by the so-called Airy distribution [29,30].…”

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confidence: 99%

Theory of Fractional Lévy Kinetics for Cold Atoms Diffusing in Optical Lattices

2012

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Recently, anomalous superdiffusion of ultracold 87Rb atoms in an optical lattice has been observed along with a fat-tailed, Lévy type, spatial distribution. The anomalous exponents were found to depend on the depth of the optical potential. We find, within the framework of the semiclassical theory of Sisyphus cooling, three distinct phases of the dynamics as the optical potential depth is lowered: normal diffusion; Lévy diffusion; and x∼t(3/2) scaling, the latter related to Obukhov's model (1959) of turbulence. The process can be formulated as a Lévy walk, with strong correlations between the length and duration of the excursions. We derive a fractional diffusion equation describing the atomic cloud, and the corresponding anomalous diffusion coefficient.

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A bernoulli excursion and its various applications

Cited by 56 publications

References 21 publications

Conditioned random walks and interaction-driven condensation

Conditioned random walks and interaction-driven condensation

Spatial survival probability for one-dimensional fluctuating interfaces in the steady state

Theory of Fractional Lévy Kinetics for Cold Atoms Diffusing in Optical Lattices

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