On the chemical distance for supercritical Bernoulli percolation

Abstract. We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z d with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

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“…We will use the following large-deviation result from Theorem 1.1 of Antal and Pisztora [2]: There exist constants a, ρ < ∞ such that…”

Section: Sublinearity Along Coordinate Directionsmentioning

confidence: 99%

“…(Indeed, the main other "external" ingredient of our proofs is Grimmett and Marstrand's paper [21] which lies at the core of [2] as well.) However, we find the argument using (4.6) conceptually cleaner and so we are content with the present, even though not necessarily optimal, proof.…”

Section: Sublinearity Along Coordinate Directionsmentioning

confidence: 99%

Quenched invariance principle for simple random walk on percolation clusters

Berger

Biskup²

2006

Probab. Theory Relat. Fields

191

258

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“…We begin in Sections 2 and 3 by finding a large (random) scale on which the percolation cluster has geometric properties which are close to those of R d . Since this random scale is not uniformly bounded (there will be some large regions where the cluster is badly behaved) we partition Z d into triadic cubes of different sizes such that every cube is well-connected in the sense of Antal and Pisztora [2], using a Calderón-Zygmund-type stopping time argument. In regions where this partition is rather coarse, the geometry of the cluster is less well-behaved and where it is finer, the cluster is well-connected.…”

Section: 1)mentioning

confidence: 99%

“…This partition plays an important role in the rest of the paper. For bounds on the "good event" which allows us to construct the partition, we use the important results of Pisztora [27], Penrose and Pisztora [26] and Antal and Pisztora [2]. We first recall some definitions introduced in those works.…”

Section: From This We Deduce That For Everymentioning

confidence: 99%

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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“…This type of renormalization has been mainly used in showing that the supercritical phase on Z d is well-behaved in several different senses, such as: possibly different critical points actually coincide [GrimM90], and the large-scale geometry (e.g., length of geodesics, isoperimetric and random walk properties) of the unique infinite cluster is very close to that of Z d [AnP96], [Pet08]. Ideally, one would also like to gain information about behavior at criticality; first of all, to show that critical percolation almost surely has no infinite clusters.…”

Section: Percolation Renormalization and Scale-invariant Tilingsmentioning

confidence: 99%

Scale-invariant groups

Nekrashevych¹,

Pete²

2011

Groups Geom. Dyn.

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Abstract. Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F o Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS.1; m/; the affine groups A Ë Z d , for any A Ä GL.Z; d /. However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.Mathematics Subject Classification (2010). 20E08, 20F18, 05B45, 60B99, 60K35.

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On the chemical distance for supercritical Bernoulli percolation

Abstract: We prove large deviation estimates at the correct order for the graph distance of two sites lying in the same cluster of an independent percolation process. We improve earlier results of G ärtner and Molchanov and Grimmett and Marstrand and answer affirmatively a conjecture of Kozlov.

Cited by 171 publications

References 12 publications

Quenched invariance principle for simple random walk on percolation clusters

Quenched invariance principle for simple random walk on percolation clusters

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

Scale-invariant groups

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