Local trapping for elliptic random walks in random environments in $$\mathbb {Z}^d$$

Fribergh, Alexander; Kious, Daniel

doi:10.1007/s00440-015-0646-4

Cited by 8 publications

(21 citation statements)

References 25 publications

(85 reference statements)

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“…1 . In this Section we also prove that condition (B) η * 1 is implied by condition (K) 1 , proposed in [4], which is the most general condition prior to this work. We formalize this implication in the following Proposition 3.1.…”

Section: 42)supporting

confidence: 55%

“…Condition (B) η * 1 is implied by the most general criteria for ballisticity for elliptic random walks in random environment. Fribergh and Kious proved in [4] that under conditions (E) 0 , (P ) M for M large enough and their condition (K) 1 the walk has ballistic behavior. It can be proved that condition (K) 1 implies (B) η * 1 .…”

Section: And This Can Be Contrasted With the Proposition Below Which ...mentioning

confidence: 99%

“…It was shown in [4] that if (C) 1 is not satisfied the random walk is not ballistic. The following is an open question: Does (C) 1 together with (P ) M for M ≥ 15d + 5 imply ballisiticity?…”

Section: Introductionmentioning

confidence: 99%

“…An ellipticity condition which is more general that (E ) 1 , denoted by (K) 1 , was defined in [4], where they showed that (K) 1 together with (P ) M for M ≥ 15d + 5 implies ballisticity. In this article we will introduce a computable ellipticity condition in dimension d = 2 and a simple general dimension condition, similar in spirit to condition (C) 1 , which also imply ballistic behavior.…”

Section: Introductionmentioning

confidence: 99%

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Computable criteria for ballisticity of random walks in elliptic random environment

Ramı́rez¹,

Ribeiro²

2021

Preprint

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We consider random walks in i.i.d. elliptic random environments which are not uniformly elliptic. We introduce a computable condition in dimension d = 2 and a general condition valid for dimensions d ≥ 2 expressed in terms of the exit time from a box, which ensure that local trapping would not inhibit a ballistic behavior of the random walk. Furthermore, we prove that this general condition is sharp.

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Section: 42)supporting

confidence: 55%

Section: And This Can Be Contrasted With the Proposition Below Which ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computable criteria for ballisticity of random walks in elliptic random environment

Ramı́rez¹,

Ribeiro²

2021

Preprint

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“…Theorem 5 in [10]). The question of integrability of renewal times in the weakly elliptic case was considered first by Campos, Ramírez ( [12]), then improved by Bouchet, Ramírez, Sabot ([10]), and Fribergh, Kious ( [23]). We state below the specification to Dirichlet case of Theorem 1 of [10].…”

Section: (T ) Condition and Integrability Of Renewal Timesmentioning

confidence: 99%

Random walks in Dirichlet environment: an overview

Sabot¹,

Tournier²

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on Z d where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights (α i ) i=1,...,2d , one for each direction of Z d . In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.

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A quenched functional central limit theorem for random walks in random environments under(T)γ

Bouchet

Sabot

Santos

2016

Stochastic Processes and their Applications

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We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [9] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra (T )γ condition of Sznitman we reduce the moment condition to E(τ 2 (ln τ ) 1+m ) < +∞ for m > 1 + 1/γ, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

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Local trapping for elliptic random walks in random environments in $$\mathbb {Z}^d$$

Cited by 8 publications

References 25 publications

Computable criteria for ballisticity of random walks in elliptic random environment

Computable criteria for ballisticity of random walks in elliptic random environment

Random walks in Dirichlet environment: an overview

A quenched functional central limit theorem for random walks in random environments under(T)γ

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