On the range of random walk

Jain, Naresh C.; Orey, Steven

doi:10.1007/bf02771217

Cited by 91 publications

(64 citation statements)

References 2 publications

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“…As M → ∞, also the third summand is converging to 0, uniformly in ε ≤ 1/2, which follows from the law of large numbers for the range of the random-walk, see [17] (notice that…”

Section: Using the Expansion (24) We Can Writementioning

confidence: 85%

Critical Behavior of Massless Free Field at Depinning Transition

Bolthausen¹,

Velenik

2001

Communications in Mathematical Physics

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We consider the d-dimensional massless free field localized by a δ-pinning of strength ε. We study the asymptotics of the variance of the field (when d = 2), and of the decay-rate of its 2-point function (when d ≥ 2), as ε goes to zero, for general Gaussian interactions. Physically speaking, we thus rigorously obtain the critical behavior of the transverse and longitudinal correlation lengths of the corresponding d + 1-dimensional effective interface model in a non-mean-field regime. We also describe the set of pinned sites at small ε, for a broad class of d-dimensional massless models.

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“…As M → ∞, also the third summand is converging to 0, uniformly in ε ≤ 1/2, which follows from the law of large numbers for the range of the random-walk, see [17] (notice that…”

Section: Using the Expansion (24) We Can Writementioning

confidence: 85%

Critical Behavior of Massless Free Field at Depinning Transition

Bolthausen¹,

Velenik

2001

Communications in Mathematical Physics

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“…But all planar random walks [21,22], and 1-dimensional β-stable random walks with β ≤ 1 [24] also satisfy Assumptions A and B.…”

Section: Remark 2 All Transient Random Walks Inmentioning

confidence: 99%

On the dynamics of trap models in ℤ^d

Fontes

Mathieu

2013

Proceedings of the London Mathematical Society

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We consider trap models in Z d . These are stochastic processes in a random environment as follows. The environment is given by a family τ = (τ x , x ∈ Z d ) of positive iid random variables in the basin of attraction of an α-stable law, 0 < α < 1. Given τ , our process is a continuous time Markov pure jump process, whose jump chain is a in principle generic random walk in Z d , d ≥ 1, independent of τ , and τ represents the holding time averages of the continuous time process. We may think of the sites of Z d as traps, and of τ x as the depth of trap x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under suitable scaling. The limit process is given by the jumps of a certain α-stable subordinator at the inverse of another α-stable subordinator, correlated with the first subordinator. For that result, the requirements on the underlying random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all random walks in d ≥ 2, and also many one dimensional random walks, but not the simple symmetric case. We then derive aging results for our process, namely scaling limits for some two-time correlation functions of the process; a strong form of those results requires an assumption of transience, stronger than a, b above. The scaling limit result mentioned above is an averaged result with respect to the environment. Under an additional condition on the size of the intersection of the ranges of two independent copies of the underlying random walk, roughly saying that it is small compared with size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to the environment. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to the environment.

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“…Strong transience is important in the theory of the range of random walks. See Jain and Pruitt [15]. After Sato [29], [30], the set T has been studied in detail by Sato and Watanabe [33], [34].…”

Section: Applications To Stable Processesmentioning

confidence: 99%

Asymptotic estimates of multi-dimensional stable densities and their applications

Watanabe

2007

Trans. Amer. Math. Soc.

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Abstract. The relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed. In particular, the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of Pruitt and Taylor in 1969 are solved. The proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere, or discrete with the support being a finite set, or a mixture of such cases are obtained. Those results are applied to the moment of the last exit time from a ball and the Spitzer type limit theorem involving capacity for a multi-dimensional transient stable process.

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On the range of random walk

Cited by 91 publications

References 2 publications

Critical Behavior of Massless Free Field at Depinning Transition

Critical Behavior of Massless Free Field at Depinning Transition

On the dynamics of trap models in ℤ^d

Asymptotic estimates of multi-dimensional stable densities and their applications

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On the range of random walk

Cited by 91 publications

References 2 publications

Critical Behavior of Massless Free Field at Depinning Transition

Critical Behavior of Massless Free Field at Depinning Transition

On the dynamics of trap models in ℤ d

Asymptotic estimates of multi-dimensional stable densities and their applications

Contact Info

Product

Resources

About

On the dynamics of trap models in ℤ^d