Long-Range Percolation Mixing Time

Benjamini, Itaı; Berger, Noam; Yadin, Ariel

doi:10.1017/s0963548308008948

Cited by 18 publications

(32 citation statements)

References 17 publications

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“…Although arising from questions in mathematical physics , the problem was recognized quickly to pose interesting challenges for probability . More recently, instances of long‐range percolation have been used as an ambient medium for other stochastic processes (e.g., ). The overarching theme here is the geometry of random networks.…”

Section: Introductionmentioning

confidence: 99%

Sharp asymptotic for the chemical distance in long‐range percolation

Biskup

Lin²

2019

Random Struct Algorithms

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We consider instances of long-range percolation on Z and R , where points at distance r get connected by an edge with probability proportional to r −s , for s ∈ ( , 2 ), and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance D(x, y) between x and y in the limit as |x − y| → ∞. For the model on Z we show that, in probability as |x| → ∞, the distance D(0, x) is squeezed between two positive multiples of (log r) Δ , where Δ ∶= 1∕ log 2 (1∕ ) for ∶= s∕(2 ). For the model on R we show that D(0, xr) is, in probability as r → ∞ for any nonzero x ∈ R , asymptotic to (r)(log r) Δ for a positive, continuous (deterministic) function obeying (r ) = (r) for all r > 1. The proof of the asymptotic scaling is based on a subadditive argument along a continuum of doubly-exponential sequences of scales. The results strengthen considerably the conclusions obtained earlier by the first author. Still, significant open questions remain.

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Section: Introductionmentioning

confidence: 99%

Sharp asymptotic for the chemical distance in long‐range percolation

Biskup

Lin²

2019

Random Struct Algorithms

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“…In [1], the discontinuity of the percolation density at s = 2 is shown. In the recent study in [8] , long range percolation mixing time is considered and it is shown that the order of the mixing time changes discontinuously when s = 2. In [18], estimates of effective resistance are given for general d, s, and the discontinuity mentioned in (1) is shown in a sense of effective resistance.…”

Section: Framework and Main Resultsmentioning

confidence: 99%

Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

2008

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We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in [6, Section 1,2], and in particular, imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long range percolation cluster. Key Words: Random walk -Random media -Heat kernel estimates -Spectral dimension -Long range percolation Running Head: Heat kernel estimates on random mediaRecently, there are intensive study for detailed properties of random walk on a percolation cluster. For a random walk on a supercritical percolation cluster on Z d (d ≥ 2), detailed Gaussian heat kernel estimates and quenched invariance principle have been obtained ([4, 10, 17, 19]). This means, such a random walk behaves in a diffusive fashion similar to a random walk on Z d . On the other hand, it is generally believed that random walk on a large critical cluster behaves subdiffusively (see [3] and the references therein). Critical percolation clusters are believed (and for some cases proved) to be finite. So, it is natural to consider random walk on an incipient infinite cluster (IIC), namely a critical percolation cluster conditioned to be infinite. Random walk on IICs has been proved to be subdiffusive on Z 2 ([13]), on trees ([14, 7]), and for the spread-out oriented percolation on Z d × Z + in dimension d > 6 ([6]).In order to study detailed properties of the random walk, it is nice and useful if one can compute the long time behaviour of the transition density (heat kernel). Let p n (x, y) be its transition density

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“…Lemma 8. 5 We have that, Proof. It is enough to prove the converge of 1 t N A,M t for a single pair (A, M) with the extension to uniform convergence over all pairs following by discretising the space.…”

Section: Lemma 81mentioning

confidence: 97%

“…This paper crucially makes use of the transience results established therein. Benjamini, Berger and Yadin [5] study the spectral gap τ of SRW on Z/NZ, providing bounds of the form cN s−1 ≤ τ ≤ CN s−1 log δ N, in that case that nearest neighbor connections exist with probability 1.…”

Section: Introductionmentioning

confidence: 99%

Simple random walk on long-range percolation clusters II: Scaling limits

Crawford¹,

Sly²

2013

Ann. Probab.

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We study limit laws for simple random walks on supercritical long range percolation clusters on Z d , d ≥ 1. For the long range percolation model, the probability that two vertices x, y are connected behaves asymptotically as x − y −s 2 . When s ∈ (d, d + 1), we prove that the scaling limit of simple random walk on the infinite component converges to an α-stable Lévy process with α = s − d establishing a conjecture of Berger and Biskup [7]. The convergence holds in both the quenched and annealed senses. In the case where d = 1 and s > 2 we show that the simple random walk converges to a Brownian motion. The proof combines heat kernel bounds from our companion paper [10], ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.

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Long-Range Percolation Mixing Time

Cited by 18 publications

References 17 publications

Sharp asymptotic for the chemical distance in long‐range percolation

Sharp asymptotic for the chemical distance in long‐range percolation

Heat Kernel Estimates for Strongly Recurrent Random Walk on Random Media

Simple random walk on long-range percolation clusters II: Scaling limits

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