Invariance principle for the random conductance model

Andrés, Sebastián; Barlow, Martin T.; Deuschel, Jean–Dominique; Hambly, Ben

doi:10.1007/s00440-012-0435-2

Cited by 81 publications

(165 citation statements)

References 39 publications

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“…It states that, P [⋅ 0 ∈ C ∞ ]-a.s., the process εX ε −2 t t≥0 converges in law, as ε → 0, to a non-degenerate Brownian motion with covariance matrix σI d . This result was extended to the setting considered here (and to even greater generality) by Biskup and Prescott [13] and Mathieu [23] (see also Andres, Barlow, Deuschel and Hambly [1]). We refer to the survey of Biskup [12] and the references therein for more on the many recent works on this problem.…”

Section: 1)supporting

confidence: 55%

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Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

Armstrong

Dario

2017

Comm Pure Appl Math

152

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Abstract. We establish quantitative homogenization, large-scale regularity and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense made precise) like that of Euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well-approximated by harmonic functions on R d . This is the first quantitative homogenization result in a porous medium and the harmonic approximation allows us to estimate the scale on which a higher-order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville-type result that states that, for each k ∈ N, the vector space of solutions growing at most like o( x k+1 ) as x → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil-Copin, Kozma, and Yadin [10] from k ≤ 1 to k ∈ N.

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Section: 1)supporting

confidence: 55%

“…Let ◻ be the largest triadic cube satisfying 1 9 ◻ ⊆ Q. It follows from simple geometric considerations that Q ⊆ ◻.…”

Section: Lemma 36 (Reverse Hölder Inequality) Fix An Exponentmentioning

confidence: 99%

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

Armstrong

Dario

2017

Comm Pure Appl Math

152

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“…(In d = 1, the decay can be arbitrarily slow.) Further progress has been made by Boukhadra [14,15] who showed that the transition from regular decay n −d/2 to anomalous decay n −2+o (1) in d ≥ 5 actually occurs in the class of power-law tails -with the exponent γ = d/2 in P(0 < ω b < s) ∼ s γ being presumably the critical for the anomaly to appear. In d ≥ 5 this meshes nicely with the annealed estimates obtained by Fontes and Mathieu [23].…”

Section: Recentmentioning

confidence: 99%

“…The principal items of interest are various asymptotics of the law of X under P x ω in the situation when ω is a sample from a probability distribution P. Much of the early effort by probabilists concerned the validity of the (functional) Central Limit Theorem. In a sequence of papers (Kipnis and Varadhan [27], De Masi, Ferarri, Goldstein and Wick [19,20], Sidoravicius and Sznitman [34], Berger and Biskup [9], Mathieu and Piatnitski [31], Mathieu [30], Biskup and Prescott [13], Barlow and Deuschel [5], Andres, Barlow, Deuschel and Hambly [1]), it has gradually been established that, as n → ∞, the law of t → X ⌊nt⌋ / √ n under P 0 ω scales to a non-degenerate Brownian motion for almost every environment ω, provided that certain conditions are met by the law of ω. where E denotes the expectation in P and p c (d) is the critical threshold for bond percolation on Z d . In d = 1 the second condition needs to be replaced by E(ω −1 b ) < ∞; independence is not required (e.g., Biskup and Prescott [13]).…”

Section: Recentmentioning

confidence: 99%

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Biskup

Boukhadra²

2012

Journal of the London Mathematical Society

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ABSTRACT. We study the diagonal heat-kernel decay for the four-dimensional nearest-neighbor random walk (on Z 4 ) among i.i.d. random conductances that are positive, bounded from above but can have arbitrarily heavy tails at zero. It has been known that the quenched return probability P 2n ω (0, 0) after 2n steps is at most C(ω)n −2 log n, but the best lower bound till now has been C(ω)n −2 . Here we will show that the log n term marks a real phenomenon by constructing an environment, for each sequence λ n → ∞, such thatwith C(ω) > 0 a.s., along a deterministic subsequence of n's. Notably, this holds simultaneously with a (non-degenerate) quenched invariance principle. As for the d ≥ 5 cases studied earlier, the source of the anomalous decay is a trapping phenomenon although the contribution is in this case collected from a whole range of spatial scales.

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“…conductances, that is when P is the product measure which includes e.g. percolation models, it is shown in [1] (building on previous works [9,10,27,28,34]) that a QFCLT holds provided that P[ω(e) > 0] > p c with p c = p c (d) being the bond percolation threshold. In particular no moment conditions such as (3) are needed.…”

mentioning

confidence: 97%

Quenched invariance principle for random walks among random degenerate conductances

Bella¹,

Schäffner²

2020

Ann. Probab.

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We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann. Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.

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Invariance principle for the random conductance model

Cited by 81 publications

References 39 publications

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models

Quenched invariance principle for random walks among random degenerate conductances

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