Asymptotic direction for random walks in random environments

Simenhaus, François

doi:10.1016/j.anihpb.2006.10.003

Cited by 11 publications

(21 citation statements)

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“…-The theorem does actually not depend on the graph structure of Z d besides translation invariance, meaning that the result also holds for non nearest neighbour models: we may enable V to be any finite subset of Z d , and the proof is written in such a way that it covers this case. The same is true for the main results of [Bo12] and [Si07] with little modification, hence the corollary also generalizes in this way in dimension ≥ 3. The intersection property for planar walks used in [ZeMe01] may however fail if jumps are allowed in such a way that the graph is not anymore planar.…”

Section: Directional Transience and Asymptotic Directionsupporting

confidence: 60%

Asymptotic direction of random walks in Dirichlet environment

Tournier¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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In this paper we generalize the result of directional transience from [SaTo10]. This enables us, by means of [Si07], [ZeMe01] and [Bo12] to conclude that, on Z d (for any dimension d), random walks in i.i.d. Dirichlet environment -or equivalently oriented-edge reinforced random walks -have almost surely an asymptotic direction equal to the direction of the initial drift, i.e. Xn Xn converges to Eo[X 1 ] Eo[X 1 ] as n → ∞, unless this drift is zero. In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of [SaTo10].P := (ω( e)) e∈V : ω( e) ≥ 0, e∈V ω( e) = 1 .

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Section: Directional Transience and Asymptotic Directionsupporting

confidence: 60%

Asymptotic direction of random walks in Dirichlet environment

Tournier¹

2015

Ann. Inst. H. Poincaré Probab. Statist.

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“…environment and ∈ S d−1 , then (P) M implies that the walk has an asymptotic direction (see [15]), i.e. the following limit…”

Section: Polynomial Condition and Atypical Quenched Exit Estimatesmentioning

confidence: 97%

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

Fribergh

Kious

2015

Probab. Theory Relat. Fields

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We consider elliptic random walks in i.i.d. random environments on Z d . The main goal of this paper is to study under which ellipticity conditions local trapping occurs. Our main result is to exhibit an ellipticity criterion for ballistic behavior which extends previously known results. We also show that if the annealed expected exit time of a unit hypercube is infinite then the walk has zero asymptotic velocity.

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“…Transience in the direction l implies that τ 1 is P 0 -a.s. finite, see [SZ99]. Due to (i), (T ) γ |l implies that condition (a) of Theorem 1 in [Sim07] is fulfilled. Hence we have the existence of an asymptotic directionv ∈ S d−1 , i.e.…”

Section: Asymptotic Direction and Atypical Quenched Exit Distributionsmentioning

confidence: 98%