Einstein relation for reversible diffusions in a random environment

Gantert, Nina; Mathieu, Pierre; Piatnitski, Andrey

doi:10.1002/cpa.20389

Cited by 25 publications

(59 citation statements)

References 28 publications

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“…By the previous observations and by the second identity in (16), we also obtain that the limit v Y (λ) := lim n→∞ Y n n (88) exists P ∞ P -a.s. and equals E[Z 0 ]v X ∞ (λ). As a consequence, v Y (λ) is deterministic, finite and strictly positive.…”

Section: Proof Of Theorem 1: the Ballistic Regimesupporting

confidence: 65%

The velocity of 1d Mott variable-range hopping with external field

Faggionato¹,

Gantert²,

Salvi³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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Mott variable range hopping is a fundamental mechanism for low-temperature electron conduction in disordered solids in the regime of Anderson localization. In a mean field approximation, it reduces to a random walk (shortly, Mott random walk) on a random marked point process with possible long-range jumps.We consider here the one-dimensional Mott random walk and we add an external field (or a bias to the right). We show that the bias makes the walk transient, and investigate its linear speed. Our main results are conditions for ballisticity (positive linear speed) and for sub-ballisticity (zero linear speed), and the existence in the ballistic regime of an invariant distribution for the environment viewed from the walker, which is mutually absolutely continuous with respect to the original law of the environment. If the point process is a renewal process, the aforementioned conditions result in a sharp criterion for ballisticity. Interestingly, the speed is not always continuous as a function of the bias.Keywords: random walk in random environment, disordered media, ballisticity, environment viewed from the walker, electron transport in disordered solids.

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Section: Proof Of Theorem 1: the Ballistic Regimesupporting

confidence: 65%

The velocity of 1d Mott variable-range hopping with external field

Faggionato¹,

Gantert²,

Salvi³

2018

Ann. Inst. H. Poincaré Probab. Statist.

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“…In our definition we follow the formulation of [SZ99]. There are two main changes from the classical regeneration time structure of the above mentioned papers: First, we have to allow the random walk to backtrack a distance of order (p − p c ) −1 , similar to the construction in [GMP12,GGN17]. Second, to obtain a stationary sequence even with backtracking, we have to control the environment where the walker regenerates.…”

Section: The Regeneration Structurementioning

confidence: 99%

Random walk on barely supercritical branching random walk

Hofstad

Hulshof

Nagel

2019

Probab. Theory Relat. Fields

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Let T be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean µ > 1, conditioned to survive. Let ϕ T be a random embedding of T into Z d according to a simple random walk step distribution. Let T p be percolation on T with parameter p, and let p c = µ −1 be the critical percolation parameter. We consider a random walk (X n ) n≥1 on T p and investigate the behavior of the embedded process ϕ Tp (X n ) as n → ∞ and simultaneously, T p becomes critical, that is, p = p n p c . We show that when we scale time by n/(p n − p c ) 3 and space by (p n − p c )/n, the process (ϕ Tp (X n )) n≥1 converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.MSC 2010. Primary: 60K37, 82C41. Secondary: 60F17, 60K40. Key words and phrases. Branching random walk, random walk indexed by a tree, percolation, scaling limit, supercriticality. arXiv:1804.04396v2 [math.PR] 13 Mar 2019 1.1. Random walk on a randomly embedded random tree. Before we proceed with the main results, let us define the model.Percolation on Galton-Watson trees. Let T be a Galton-Watson tree rooted at with law P, and let ξ be the random number of offspring of the root. Suppose that the tree is supercritical, i.e., µ := E[ξ] > 1. We write ∆ for the maximal number of children that a single individual in the (unpercolated) tree can have, i.e., ∆ := sup{n : P(ξ = n) > 0}.We consider percolation on the edges of the tree and let T p denote the connected component of the root in T , when each edge is deleted with probability 1 − p ∈ (0, 1), independently of every other edge and of T . Then T p is again a Galton-Watson tree, whose distribution we denote by P p . The root of a tree T chosen according to P p now has ξ p offspring, such that, conditioned on {ξ = n}, ξ p is a binomial random variable with parameters n and p.The percolated tree has mean number of offspring pµ and is supercritical if and only if p > p c := 1/µ. In this setting, denote byP p = P p ( · | |T | = ∞) the distribution of the percolated tree, conditioned on non-extinction. Given a tree T rooted at , let (X n ) n≥0 be a simple random walk on T started at . Given v ∈ T , we write P v T for the law of (X n ) n≥0 with X 0 = v, and write P T if v = . Given a realization of a Galton-Watson tree T , we call P T the quenched law of the random walk, and we call P p :=P p × P T the annealed law of the random walk on the tree (writing P v p :Spatial embedding: branching random walk. We embed a Galton-Watson tree T into Z d by means of a branching random walk, which we will define now: Let D denote a non-degenerate probability distribution on Z d . Given a tree T rooted at , set Z( ) = 0 and assign to each vertex v = of T an independent random variable Z(v) with law D. For any v ∈ T there exists a unique path = v 0 , v 1 , . . . , v m = v. The branching random walk embedding ϕ = ϕ T is defined such that ϕ(v) :=...

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“…It finally remains to prove (8.33). We begin by proving the analogue of Lemma 5.13 in [13]. While the proof is essentially the same, we have to replace the independence property of the times between two regenerations by the 1dependence property.…”

Section: Definition 84mentioning

confidence: 99%

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

2019

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We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0, then its asymptotic linear speed v is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk. p ,λ . If the walk starts at v = 0, we sometimes omit the superscript 0. Further, if λ = 0, we sometimes omit λ as a subscript, and write p ω for p ω,0 , and P for P 0 .

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Einstein relation for reversible diffusions in a random environment

Cited by 25 publications

References 28 publications

The velocity of 1d Mott variable-range hopping with external field

The velocity of 1d Mott variable-range hopping with external field

Random walk on barely supercritical branching random walk

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

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