Effective Polynomial Ballisticity Conditions for Random Walk in Random Environment

Berger, Noam; Drewitz, Alexander; Ramı́rez, Alejandro F.

doi:10.1002/cpa.21500

Cited by 33 publications

(94 citation statements)

References 17 publications

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“…Subsequent works ( [7,8] and [1]) have weakened the transience conditions that we can verify to prove ballistic behavior under uniform ellipticity. At this point in time, the state of the art is a result from [1] called polynomial condition typically denoted (P) M .…”

Section: Former Results and Open Questionsmentioning

confidence: 96%

“…The main result of [1] is that for a RWRE in i.i.d. environment with uniformly elliptic transition probabilities then (P) M for M ≥ 15d + 5 implies (T ).…”

Section: Former Results and Open Questionsmentioning

confidence: 99%

“…At this point in time, the state of the art is a result from [1] called polynomial condition typically denoted (P) M .…”

Section: Former Results and Open Questionsmentioning

confidence: 99%

“…However, in concrete examples (for example, see Proposition 4.3, which exhibits new examples of ballistic walks), we can use a much simpler criterion ( K ) 1 defined by: 1 is easily seen to imply (K ) 1 , by choosing the γ 's and the Markovian hypercube conveniently. Indeed, assume ( K ) 1 is verified and denote x min ∈ H the vertex for which the minimum is reached.…”

Section: Theorem 32 Let Us Consider a Rwre In An Elliptic Iid Envmentioning

confidence: 98%

“…It is expected that if the transition probabilities are uniformly elliptic then the walk is ballistic (see [20] and [23]). This conjecture has been proved under stronger transience assumptions known as Sznitman's conditions (T ), (T ) or (T ) γ (see [18] and [19]) and more recently condition (P) M (see [1] [20] and [23]). Proving this equivalence is one of the major open problems in random walk in random environments (RWRE).…”

Section: Introductionmentioning

confidence: 98%

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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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Section: Former Results and Open Questionsmentioning

confidence: 96%

“…The main result of [1] is that for a RWRE in i.i.d. environment with uniformly elliptic transition probabilities then (P) M for M ≥ 15d + 5 implies (T ).…”

Section: Former Results and Open Questionsmentioning

confidence: 99%

“…At this point in time, the state of the art is a result from [1] called polynomial condition typically denoted (P) M .…”

Section: Former Results and Open Questionsmentioning

confidence: 99%

Section: Theorem 32 Let Us Consider a Rwre In An Elliptic Iid Envmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

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

Fribergh

Kious

2015

Probab. Theory Relat. Fields

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A Proof of Sznitman's Conjecture about Ballistic RWRE

Guerra

Ramı́rez

2019

Comm Pure Appl Math

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We consider a random walk in a uniformly elliptic i.i.d. random environment in Z d for d ≥ 2. It is believed that whenever the random walk is transient in a given direction it is necessarily ballistic. In order to quantify the gap which would be needed to prove this equivalence, several ballisticity conditions have been introduced. In particular, in [Sz01, Sz02], Sznitman defined the so called conditions (T) and (T ). The first one is the requirement that certain unlikely exit probabilities from a set of slabs decay exponentially fast with their width L. The second one is the requirement that for all γ ∈ (0, 1) condition (T) γ is satisfied, which in turn is defined as the requirement that the decay is like e −CL γ for some C > 0. In this article we prove a conjecture of Sznitman of 2002 [Sz02], stating that (T) and (T ) are equivalent. Hence, this closes the circle proving the equivalence of conditions (T), (T ) and (T) γ for some γ ∈ (0, 1) as conjectured in [Sz02], and also of each of these ballisticity conditions with the polynomial condition (P) M for M ≥ 15d + 5 introduced by Berger, Drewitz and Ramírez in [BDR14].

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Selected Topics in Random Walks in Random Environment

Drewitz

Ramı́rez

2014

Topics in Percolative and Disordered Systems

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Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers as a model of DNA chain replication and crystal growth (see Chernov [Ch67] and Temkin [Te69, Te72]), and also as a model of turbulent behavior in fluids through a Lorentz gas description (Sinaȋ 1982 [Si82a]). It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology and engineering. While the one-dimensional model is well-understood, in the multidimensional setting, fundamental questions about the RWRE model have resisted repeated and persistent attempts to answer them. Two major complications in this context stem from the loss of the Markov property under the averaged measure as well as the fact that in dimensions larger than one, the RWRE is not reversible anymore. In these notes we present a general overview of the model, with an emphasis on the multidimensional setting and a more detailed description of recent progress around ballisticity questions.We will usually call the environment (uniformly) elliptic in that case also.Remark 2.4. This labeling is motivated by operator theory where one has analogous definitions of elliptic and uniformly elliptic differential operators.The following auxiliary process will play a significant role in what follows.Definition 2.5. (Environment viewed from the particle). Let (X n ) be a RWRE. We define the environment viewed from the particle (or also the environmental process) as the discrete time processω n := t Xn ω, for n ≥ 0, with state space Ω.Apart from taking values in a compact state space, another advantage of the environment viewed from the particle is that even under the averaged measure it is Markovian, as is shown in the next result following Sznitman [BS02]; however, the cost is that we now deal with an infinite dimensional state space.Proposition 2.6. Consider a RWRE in an environment with law P. Then, under P 0 , the process (ω n ) is Markovian with state space Ω, initial law P, and transition kernel Rf (ω) := e∈U ω(0, e)f (t e ω), (2.8) defined for f bounded measurable on Ω and initial law P. Proof. Let us first note that for every x ∈ Z d , and every bounded measurable function f on Ω, E x,ω [f (ω 1 )] = E x,ω [f (t X 1 ω)] = e∈U ω(x, e)f (t x+e ω) = e∈U t x ω(0, e)f (t e (t x ω)) = Rf (t x ω).(2.9)

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Effective Polynomial Ballisticity Conditions for Random Walk in Random Environment

Cited by 33 publications

References 17 publications

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

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

A Proof of Sznitman's Conjecture about Ballistic RWRE

Selected Topics in Random Walks in Random Environment

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