Regularity of the time constant for a supercritical Bernoulli percolation

Dembin, Barbara

doi:10.1051/ps/2021005

Cited by 10 publications

(4 citation statements)

References 17 publications

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“…This model is equivalent to allow t(e) to be +∞ in first-passage percolation on L d . In the special case where P ρ (t(e) = 1) = ρ and P ρ (t(e) = +∞) = 1 − ρ, the regularity of the time constant and the limit shape with respect to ρ ∈ (p c , 1] is explored by Dembin [8] and Cerf and Dembin [3]. It is shown in [3] that for p 0 > p c , the time constant is Lipschitz continuous on ρ ∈ [p 0 , 1].…”

Section: Resultsmentioning

confidence: 99%

Comparison of limit shapes for Bernoulli first-passage percolation

Naoki¹,

Takei²

2022

Preprint

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We consider Bernoulli first-passage percolation on the ddimensional hypercubic lattice with d ≥ 2. The passage time of edge e is 0 with probability p and 1 with probability 1−p, independently of each other. Let pc be the critical probability for percolation of edges with passage time 0. When 0 ≤ p < pc, there exists a nonrandom, nonempty compact convex set Bp such that the set of vertices to which the firstpassage time from the origin is within t is well-approximated by tBp for all large t, with probability one. The aim of this paper is to prove that for 0 ≤ p < q < pc, the Hausdorff distance between Bp and Bq grows linearly in q − p. Moreover, we mention that the approach taken in the paper provides a lower bound for the expected size of the intersection of geodesics, that gives a nontrivial consequence for the critical case.

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Section: Resultsmentioning

confidence: 99%

Comparison of limit shapes for Bernoulli first-passage percolation

Naoki¹,

Takei²

2022

Preprint

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“…This problem is, in turn, closely related to the problem of computing the asymptotics of the time constant for supercritical percolation as p ↓ p c . An analogous problem for high-dimensional random interlacements has recently been solved to within subpolynomial factors in [17], and regularity results for the percolation time constant have been established in [6,9,10].…”

Section: Expected Volume Growthmentioning

confidence: 99%

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

Hutchcroft¹

2022

Preprint

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We study the growth and isoperimetry of infinite clusters in slightly supercritical Bernoulli bond percolation on transitive nonamenable graphs under the L 2 boundedness condition (p c < p 2→2 ). Surprisingly, we find that the volume growth of infinite clusters is always purely exponential (that is, the subexponential corrections to growth are bounded) in the regime p c < p < p 2→2 , even when the ambient graph has unbounded corrections to exponential growth. For p slightly larger than p c , we establish the precise estimatesfor every v ∈ V , r ≥ 0, and p c < p ≤ p c + δ, where the growth rate γ int (p) = lim 1 r log E p #B(v, r) satisfies γ int (p) ≍ p − p c . We also prove a percolation analogue of the Kesten-Stigum theorem that holds in the entire supercritical regime and states that the quenched and annealed exponential growth rates of an infinite cluster always coincide. We apply these results together with those of the first paper in this series to prove that the anchored Cheeger constant of every infinite cluster K satisfiesalmost surely for every p c < p ≤ 1.

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“…In fact, there is not much research related to our topic. In [1], [2], and [3] the Lipschitz continuity is obtained for the so-called time constant of the first passage percolation on and the isoperimetric constant of the supercritical percolation cluster (which are counterparts of the Lyapunov exponent), respectively. The aim of [17] is to derive a (non-trivial) lower bound for the difference between the time constants.…”

Section: Introductionmentioning

confidence: 99%

Differences between Lyapunov exponents for the simple random walk in Bernoulli potentials

Kubota

2023

J. Appl. Probab.

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We consider the simple random walk on the d-dimensional lattice $\mathbb{Z}^d$ ( $d \geq 1$ ), traveling in potentials which are Bernoulli-distributed. The so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential, and it is known that the Lyapunov exponent is strictly monotone in the parameter of the Bernoulli distribution. Hence the aim of this paper is to investigate the effect of the potential on the Lyapunov exponent more precisely, and we derive some Lipschitz-type estimates for the difference between the Lyapunov exponents.

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Regularity of the time constant for a supercritical Bernoulli percolation

Cited by 10 publications

References 17 publications

Comparison of limit shapes for Bernoulli first-passage percolation

Comparison of limit shapes for Bernoulli first-passage percolation

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

Differences between Lyapunov exponents for the simple random walk in Bernoulli potentials

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