Isoperimetry and heat kernel decay on percolation clusters

By Pierre Mathieu, É. Remy. 2003

139 citations recorded by the source corpus.

We prove that the heat kernel on the infinite Bernoulli percolation cluster in \Zd almost surely decays faster than t−d/2. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities. Some of the results of this paper were previously announced in the note of Mathieu and Remy [C. R. Acad. Sci. Paris Ser. I Math. 332 (2001) 927--931].

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0cda6549fca6e33b2c292c2911271bfd1682f896

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Citations in the corpus

Random walks on supercritical percolation clusters (2003, methodology): “2 of [MR] gives the (quenched) bound q t (0, y) ≤ c1(p, d)t , t ≥ T0(ω), y ∈ C∞, Pp-a.”
On symmetric random walks with random conductances on ℤd (2004, methodology): “Calling P ω x [X N s = y] the transition probabilities for the random walk on Cω N , we quote from formula (6) of [6]: Q-a.”
Quenched invariance principle for random walks with time-dependent ergodic degenerate weights (2016, unknown): “There exists θ ∈ ( 0 , 1 ) and N(x) < ∞ , P -a.s., such that for all n ≥ N(x) , for all connected A ⊂ B(x, n) with | A | ≥ n θ ; see [11, 31].”
Invariance principle for the random conductance model in a degenerate ergodic environment (2013, background): “In the particular case of supercritical percolation clusters on Z , a stronger isoperimetric inequality has been proven in [32].”
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].”
Quenched invariance principles for random walks on percolation clusters (2005, background): “In case t%jxKyj%2t, then (2.3) follows from the Carne–Varopoulos bound (see appendix C in Mathieu & Remy (2004)).”
Recent progress on the Random Conductance Model (2011, background): “Benjamini and Mossel [10], Heicklen and Hoffman [78], Mathieu and Remy [102], Barlow [5], Berger-Biskup-Hoffman-Kozma [12], Pete [115].”
Gaussian Upper Bounds for Heat Kernels of Continuous Time Simple Random Walks (2011, background): “In [21], Mathieu and Remy proved the following on-diagonal heat kernel bound for the CSRW on Cp,∞(ω).”
A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion (2007, methodology): “Assume 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,…”
The discrete integral maximum principle and its applications (2005, unknown): “(see [20] and also [ 21 ] for a continuous time analogue, and [4] for full Gaussian upper and lower bounds for a continuous time random walk on percolation clusters).”
Evolving sets, mixing and heat kernel bounds (2003, result): “(See [17] for analogous results in higher dimensions.”
Quenched invariance principle for simple random walk on percolation clusters (2005, methodology): “(In fact, the continuous-time version of the bound (6.14) was obtained already by Mathieu and Remy [31].)”