Random walk in a fluctuating random environment with Markov evolution

Boldrighini, C.; Minlos, R. A.; Пеллегринотти, Алессандро

doi:10.1090/trans2/198/02

Cited by 20 publications

(22 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The main term of the asymptotic estimate P (X T = y | X 0 = 0, ξ) as T → ∞ does not depend on ξ and has the same parameters as the average random walk. For ν ≥ 3 an analogous result was found in the Markov case ( [BMP3]). …”

Section: Introductionsupporting

confidence: 77%

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

Bernabei¹

2001

Probab Theory Relat Fields

View full text Add to dashboard Cite

The Central Limit Theorem for a model of discrete-time random walks on the lattice ‫ޚ‬ ν in a fluctuating random environment was proved for almost-all realizations of the space-time environment, for all ν > 1 in [BMP1] and for all ν ≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν ≥ 3 is finite. In the present paper we consider the cases ν = 1, 2 and prove the Central Limit Theorem as T → ∞ for the random correction to the first two cumulants. The rescaling factor for the average is T 1 4 for ν = 1 and (ln T ) 1 2 , for ν = 2; for the covariance it is T 1− ν 4 , ν = 1, 2. IntroductionIn this paper we extend the main result of [BMP1] and [BBMP]. We consider a natural class of models of discrete-time random walks in random media on the lattice ‫ޚ‬ ν , ν = 1, 2, . . ., which evolve stochastically in time t. The environment is described by a field ξ = {ξ t (x) : x ∈ ‫ޚ‬ ν , t ∈ ‫ޚ‬ + } of random variables taking values in some finite set S. We denote by X t the particle position at time t.The random walk transition probabilities at time t iswhere P 0 is a probability distribution of an homogeneous random walk, and ε a small parameter. The function c represents the influence of the field on the evolution of the random walk. The interaction is "strictly local", i.e. confined to the place occupied by the particle at time t. The functions P 0 and c have compact support ("finite range" assumption).In general one assumes that the {ξ t (x), t ≥ 0}, x ∈ ‫ޚ‬ ν , are independent copies of an ergodic Markov chain. One can suppose a further local interaction of the

show abstract

Section: Introductionsupporting

confidence: 77%

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

Bernabei¹

2001

Probab Theory Relat Fields

View full text Add to dashboard Cite

show abstract

“…In fact for ν 3, this has been proved in [6], and at present the results in low dimension ν = 1, 2 are available as well [8].…”

Section: Introductionmentioning

confidence: 62%

“…Almost-sure asymptotics requires accurate estimates of the correction terms. This is done in [6], where the quenched CLT is proved for ν 3, almost surely with respect to ℘ Π , where Π = π Z ν is the equilibrium measure for the environment. The result is as follows.…”

Section: Quenched Random Walkmentioning

confidence: 99%

“…The proof in [6] relies on the construction of an explicit orthonormal basis in the space of the trajectories of the single site Markov chain and on cluster expansion methods.…”

Section: Quenched Random Walkmentioning

confidence: 99%

“…ii) There are positive constants C F and q ∈ (0, 1), C F depending only on F and q independent of F and Γ such that 6) where d(Γ, 0) is the distance of the set Γ from the origin in Z ν .…”

Section: Environment From the Point Of View (Epv)mentioning

confidence: 99%

See 2 more Smart Citations

Random walks in random environment with Markov dependence on time

Boldrighini¹,

Minlos²,

Pellegrinotti³

2008

Condens. Matter Phys.

View full text Add to dashboard Cite

We consider a simple model of discrete-time random walk on Z ν , ν = 1, 2, . . . in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.

show abstract

Random Walks in Random Environment

Zeitouni¹

2009

Encyclopedia of Complexity and Systems Science

View full text Add to dashboard Cite

Random walks in random environments (RWRE's) have been a source of surprising phenomena and challenging problems since they began to be studied in the 70's. Hitting times and, more recently, certain regeneration structures, have played a major role in our understanding of RWRE's. We review these and provide some hints on current research directions and challenges.2000 Mathematics Subject Classification: 60K37, 82C44.

show abstract

Random walk in a fluctuating random environment with Markov evolution

Cited by 20 publications

References 0 publications

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

Random walks in random environment with Markov dependence on time

Random Walks in Random Environment

Contact Info

Product

Resources

About