Quenched invariance principles for walks on clusters of percolation or among random conductances

By Vladas Sidoravicius, A. Sznitman. 2004

220 citations recorded by the source corpus.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7b048e7716ed4cd53c8108bd0294db20aca68f6d

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

A Regularity Theory for Random Elliptic Operators (2014, background): “− Br (φ,σ)|2 = 0 for a. e. a. For φ this statement (in a more involved form) is a key ingredient for the quenched invariance principle and can be established based on ergodicity and stationarity, see [60]. We argue in Step 1 that the same argument can be used to establish this property for σ. 32 A. ”
A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion (2007, background): “…< ∞, (4.1) which quantity does not depend on the basepoint o. (4.1) is known to be satisfied in many situations: when G is the Cayley graph of a finitely presented group that is not a finite extension of Z, or is quasi-isometric to such a Cayley graph [3, 43]; when G is a planar graph with…”
Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “We will proceed as in [36], pages 229–230.”
Quenched invariance principle for random walks with time-dependent ergodic degenerate weights (2016, background): “ N we have [−n,n]d ∩Zd ⊂ B(2n) ⊂ [−2n,2n]d ∩Zd, it sufﬁces to prove the lemma for boxes in place of balls. STEP 1: We show the existence of the limit similarly to [5, Lemma 2.8] based on arguments in [36]. By symmetry it sufﬁces to show the existence of lim n→∞ 1 nd+1 X x∈C(n) Fn(x)χ(ω,0,x), where C”
The Alexander-Orbach conjecture holds in high dimensions (2008, background): “r a comprehensive exposition). In particular, the random walk on the supercritical inﬁnite cluster has an invariance principle, the spectral dimension is ds = d and other Zd-like properties hold, see [19, 13, 5, 52, 15, 43]. Our analysis establishes that balls of radius r in the IIC typically have v”
A central limit theorem for biased random walks on Galton–Watson trees (2006, background): “This martingale provides “harmonic coordinates” for the random walk, in the spirit of [13] and, more recently, [5,22].”
Quenched invariance principles for random walks on percolation clusters (2005, background): “We refer in particular to the first part of Sidoravicius & Sznitman (2004), in which random walks in elliptic environments are considered.”
Quenched invariance principle for random walks in balanced random environment (2010, background): “Much of this work has been in the context of reversible models, either for walks on percolation clusters or for the random conductance model, see [1, 19, 15, 4, 16, 14, 2].”