Comparison of quenched and annealed invariance principles for random conductance model

Barlow, Martin T.; Burdzy, Krzysztof; Timár, Ádám

doi:10.1007/s00440-015-0618-8

Cited by 20 publications

(53 citation statements)

References 17 publications

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“…Recall that in view of the counterexample in [11] a moment condition is necessary already for the (non-quantitative) invariance principle to hold.…”

Section: Assumption 13 (Spectral Gap)mentioning

confidence: 99%

“…However, in the case of a general ergodic environment, due to a trapping phenomenon, it is clear that some moment conditions (stronger than the one in (1.3)) are needed. Indeed, Barlow, Burdzy and Timar [11] give an example on Z 2 which satisfies a weak moment condition, but not a quenched invariance principle. To formulate the moment condition used in our paper, we set for any p, q ∈ [1, ∞],…”

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confidence: 99%

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Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Andrés

Neukamm

2018

Stoch PDE: Anal Comp

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We study the random conductance model on the lattice Z d , i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension d ≥ 3 quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed t − 1 5 +ε for d ≥ 4 and t − 1 10 +ε for d = 3. Additionally, in the uniformly elliptic case in low dimensions d = 2, 3 we improve the rate in a quantitative Berry-Esseen theorem recently obtained by Mourrat. As a central analytic ingredient, for d ≥ 3 we establish near-optimal decay estimates on the semigroup associated with the environment process. These estimates also play a central role in quantitative stochastic homogenization and extend some recent results by Gloria, Otto and the second author to the degenerate elliptic case.1.1. The model. Let d ≥ 2. We study the nearest-neighbour random conductance model on the d-dimensional Euclidean lattice (Z d , E d ), where E d := {e = {x, x ± e i } : x ∈ Z d , i = 1, . . . , d } denotes the set of non-oriented nearest neighbour edges and {e 1 , . . . , e d } the canonical basis in R d . We endow the graph with positive random weights, which we describe by a family ω = {ω(e), e ∈ E d } ∈ Ω := (0, ∞) E d . We refer to ω(e) as the conductance of an edge e ∈ E d . To simplify notation, for any x, y ∈ Z d , we set ω(x, y) = ω(y, x) := ω({x, y}), ∀ {x, y} ∈ E d , ω(x, y) := 0, ∀ {x, y} ∈ E d , and define the matrix field ω : Z d → R d×d by ω(x) = diag(ω(x, x + e 1 ), . . . , ω(x, x + e d )). 2 1 5 if d = 2, c (t + 1) − 1 5 if d ≥ 3.5

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“…Recall that in view of the counterexample in [11] a moment condition is necessary already for the (non-quantitative) invariance principle to hold.…”

Section: Assumption 13 (Spectral Gap)mentioning

confidence: 99%

mentioning

confidence: 99%

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Andrés

Neukamm

2018

Stoch PDE: Anal Comp

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“…To prove Theorem 1 note that by Lemmas 6 and 7 it suffices to find C, γ, δ > 0 such that (2) P 0 (T > n) ≤ Ce −γn δ , for every n.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For any 2-valued model µ(γ (1) ) = p = 1 − µ(γ (2) ) ∈ (0, 1), let N (i) n = #{0 ≤ m < n : ω Xm = γ (i) } and note that N (1) n + N (2) n = n. Let us write N n for N (1) n .…”

Section: Proof Of Propositionmentioning

confidence: 99%

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Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

Holmes¹,

Salisbury²

2017

Electron. J. Probab.

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We study the asymptotic behaviour of random walks in i.i.d. non-elliptic random environments on Z d . Standard conditions (and proofs) for ballisticity and the central limit theorem require ellipticity. We use oriented percolation and martingale arguments to give non-trivial local conditions for ballisticity and an annealed invariance principle in the non-elliptic setting.2000 Mathematics Subject Classification. 60K37.

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Conduction and Diffusion in Percolating Systems

Hughes¹

2021

Complex Media and Percolation Theory

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Comparison of quenched and annealed invariance principles for random conductance model

Abstract: We show that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold.

Cited by 20 publications

References 17 publications

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

Conditions for ballisticity and invariance principle for random walk in non-elliptic random environment

Conduction and Diffusion in Percolating Systems

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