First occurrence time of a large density fluctuation for a system of independent random walks (original) (raw)
A special set of exceptional times for dynamical random walk on Z 2
Electronic Journal of Probability}
Benjamini, Häggström, Peres and Steif introduced the model of dynamical random walk on d [2]. This is a continuum of random walks indexed by a parameter t. They proved that for d = 3, 4 there almost surely exist t such that the random walk at time t visits the origin infinitely ...
Limit theorems for Levy walks in d dimensions: rare and bulk fluctuations
Journal of Physics A: Mathematical and Theoretical, 2017
We consider super-diffusive Lévy walks in d 2 dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d-dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d-dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the d-dimensional process can be reduced to one-dimensional Lévy walk.
Stable fluctuations for ballistic random walks in random environment on Z
arXiv (Cornell University), 2010
We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level n, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following [7], where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. Like in [7], the case of Dirichlet environment turns out to be remarkably explicit.
Limit theorems for Lévy walks inddimensions: rare and bulk fluctuations
Journal of Physics A, 2017
We consider super-diffusive Lévy walks in d 2 dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d−dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d−dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the d−dimensional process can be reduced to one-dimensional Lévy walk.
The Annals of Applied Probability, 2013
We consider transient nearest-neighbor random walks in random environment on Z. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level n, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975) 145-168].
Scaling property of flux fluctuations from random walks
Physical Review E, 2007
We study dynamical scaling of flux fluctuation ͑t͒ from the one-random-walker model on regular lattices and complex networks and compare it to the surface width W͑t͒ of a corresponding growth model. On the regular lattices, we analytically show that ͑t͒ undergoes a crossover from the nontrivial scaling regime to the trivial one by increasing time t, and we verify the results by numerical simulations. In contrast to the results on the regular lattices, ͑t͒ does not show any crossover behavior on complex networks and satisfies the scaling relation ͑t͒ϳt 1/2 for any t. On the other hand, we show that W͑t͒ of the corresponding model on complex networks has two different scaling regimes, W ϳ t 1/2 for t Ӷ N and W͑t͒ϳt for t ӷ N.
Annealed and quenched fluctuations for ballistic random walks in random environment on Z
arXiv (Cornell University), 2010
We consider transient random walks in random environment on Z\ZZ in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level nnn, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following [7], where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. Furthermore, our proof enables us to give a description of the quenched distribution of hitting times. The case of Dirichlet environment turns out to be remarkably explicit.
Infinite densities in continuous-time random walks
We study occupation time statistics in ergodic continuous-time random walks. Under thermal detailed balance conditions, the average occupation time is given by the Boltzmann-Gibbs canonical law. The finite-time fluctuations around this mean turn out to exhibit dual time scaling and distribution laws. The infinite density of large fluctuations complements the Lévy-stable density of bulk fluctuations. Neither of the two should be interpreted as a stand-alone limiting law, as each has its own deficiency: the infinite density has an infinite norm (despite particle conservation), while the stable distribution has an infinite variance (although occupation times are bounded). These unphysical divergences are remedied by consistent use and interpretation of both formulas. Interestingly, while the system's canonical equilibrium laws naturally determine the mean occupation time of the ergodic motion, they also control the infinite and stable densities of fluctuations.
Limit laws for transient random walks in random environment on z\zz
HAL (Le Centre pour la Communication Scientifique Directe), 2009
We consider transient random walks in random environment on Z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level n converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.
On the behavior of random walk around heavy points
2006
Consider a symmetric aperiodic random walk in ZdZ^dZd, dgeq3d\geq 3dgeq3. There are points (called heavy points) where the number of visits by the random walk is close to its maximum. We investigate the local times around these heavy points and show that they converge to a deterministic limit as the number of steps tends to infinity.