Einstein relation for reversible random walks in random environment on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:math>

Lam, Hoang-Chuong; Depauw, Jérôme

doi:10.1016/j.spa.2015.10.007

Cited by 3 publications

(9 citation statements)

References 9 publications

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“…negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of [25].…”

supporting

confidence: 71%

“…We prove the Einstein relation under two different sets of assumptions: the first one requires the ergodicity of the conductances and some moments conditions; we point out that this is the equivalent set of conditions that [25] would require in our setting, but we give an alternative, shorter proof. We extend this result including a different hypothesis, just requiring the weakest possible integrability of the conductances and very mild mixing conditions.…”

Section: Einstein Relationmentioning

confidence: 99%

“…Call F k the σ-algebra generated by (θ iS (λ)) i≤k . In the proof of Theorem 2.2.1 in [34] one only needs inequality (25) with G −n replaced by F −n . This is indeed automatically satisfied when (25) holds since F k ⊂ G k and therefore, for each random variable Z,…”

Section: Appendix B Proof Of Proposition 41mentioning

confidence: 99%

“…In this paper we investigate the behavior of the quantities v(λ) and σ 2 (λ) as functions of the parameter λ. In particular, we focus on their monotonicity, differentiability and analyticity, and we derive the Einstein Relation for the random conductance model (RCM) extending the result of [25]. The advantage of working with nearest neighbor walks on the one-dimensional lattice is that v(λ) and σ 2 (λ) have an explicit representation in terms of suitable series (see [34]).…”

Section: Introductionmentioning

confidence: 99%

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Regularity of biased 1D random walks in random environment

Faggionato¹,

Salvi²

2019

ALEA

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We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ ∈ R. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+. When λ− and λ+ are distinct, v(λ) might fail to be continuous. We refine the assumptions in [34] for having a recentered CLT with diffusivity σ 2 (λ) and give explicit conditions for σ 2 (λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ 2 (λ) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of [25].

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“…negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of [25].…”

supporting

confidence: 71%

Section: Einstein Relationmentioning

confidence: 99%

Section: Appendix B Proof Of Proposition 41mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularity of biased 1D random walks in random environment

Faggionato¹,

Salvi²

2019

ALEA

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“…A weaker form of the Einstein relation, which is often used as a starting point, was proved in [29]. Since then, the analysis of the Einstein relation, the steady states and the linear response for random walks in static/dynamic random environments have been addressed in [2,3,17,18,19,23,24,27,28,30,31,33,35] (the list is not exhaustive). The approach used here is different from the previous works: Although the distribution Q λ is not explicit, by refining the analysis of [12] we prove that the Radon-Nikodym derivative dQ λ dQ 0 belongs to L p (Q 0 ) if E e pZ 0 < ∞ for some p ≥ 2 (see Theorem 1).…”

Section: Introductionmentioning

confidence: 99%

Einstein relation and linear response in one-dimensional Mott variable-range hopping

Faggionato¹,

Gantert²,

Salvi³

2019

Ann. Inst. H. Poincaré Probab. Statist.

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We consider one-dimensional Mott variable-range hopping with a bias, and prove the linear response as well as the Einstein relation, under an assumption on the exponential moments of the distances between neighboring points. In a previous paper [12] we gave conditions on ballisticity, and proved that in the ballistic case the environment viewed from the particle approaches, for almost any initial environment, a given steady state which is absolutely continuous with respect to the original law of the environment. Here, we show that this bias-dependent steady state has a derivative at zero in terms of the bias (linear response), and use this result to get the Einstein relation. Our approach is new: instead of using e.g. perturbation theory or regeneration times, we show that the Radon-Nikodym derivative of the bias-dependent steady state with respect to the equilibrium state in the unbiased case satisfies an L p -bound, p > 2, uniformly for small bias. This L p -bound yields, by a general argument not involving our specific model, the statement about the linear response.

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