Random walks on supercritical percolation clusters

By Martin T. Barlow. 2003

248 citations recorded by the source corpus.

We obtain Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x; ) only holds for t Sx(!), where the constant Sx(!) depends on the percolation congura- tion !.

Subfields: Geometry and Topology, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fa36745009d6646d3a98e03970eef6bce8c6c722

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Citations in the corpus

Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “[5, 21, 32]) have been used so far in the proofs of the quenched functional CLT; cf.”
Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks (2011, background): “For existing work on random walks on percolation clusters, including on-diagonal heat kernel estimates and invariance principles, see [21] and [3].”
The Alexander-Orbach conjecture holds in high dimensions (2008, background): “In particular, the random walk on the supercritical infinite cluster has an invariance principle, the spectral dimension is ds = d and other Z d-like properties hold, see [19, 13, 5, 49, 15, 42].”
Recent progress on the Random Conductance Model (2011, background): “(5.14) There have been a number of proofs of this and/or related results, see e.g. Benjamini and Mossel [10], Heicklen and Hoffman [78], Mathieu and Remy [102], Barlow [5], Berger-Biskup-Hoffman-Kozma [12], Pete [115].”
A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion (2007, background): “…that κ(G) := lim sup n→∞ qn(G) 1/n < ∞, (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];…”
On symmetric random walks with random conductances on ℤd (2004, background): “Estimates on the return probability for random walks on percolation clusters have been proved in [6] (See also [1] ).”
Quenched invariance principles for random walks on percolation clusters (2005, methodology): “For percolation models, a weaker but still suitable form of the Poincaré inequality was proved by Mathieu & Remy (2004) (see also Barlow 2004).”
Quenched invariance principle for random walks in balanced random environment (2010, methodology): “Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.”
Quenched invariance principles for walks on clusters of percolation or among random conductances (2004, methodology): “We will make repeated use of Gaussian bounds which for the purpose of this section provide more than enough control on the walk, see Theorem 1.1 and subsequent remarks in Barlow [ 3 ], see also [25] and [16]: Quenched invariance principles for walks on clusters of percolation 231”
Harnack inequalities on weighted graphs and some applications to the random conductance model (2013, background): “We prove here a strong version of the local Poincaré inequality in the sense that the balls on both sides of inequality have the same radius, in contrast to weak local Poincaré inequalities used for instance in [6].”