## ON THE CRITICAL PARAMETER OF INTERLACEMENT PERCOLATION IN HIGH DIMENSION (2010)

Citations: | 2 - 2 self |

### BibTeX

@MISC{Sznitman10onthe,

author = {Alain-sol Sznitman},

title = {ON THE CRITICAL PARAMETER OF INTERLACEMENT PERCOLATION IN HIGH DIMENSION},

year = {2010}

}

### OpenURL

### Abstract

The vacant set of random interlacements on Z d, d ≥ 3, has non-trivial percolative properties. It is known from [18], [16], that there is a non-degenerate critical value u ∗ such that the vacant set at level u percolates when u < u ∗ and does not percolate when u> u∗. We derive here an asymptotic upper bound on u∗, as d goes to infinity, which complements the lower bound from [21]. Our main result shows that u ∗ is equivalent to log d for large d, and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in [23].

### Citations

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Citation Context ...erm of (1.9) is 2t smaller than (1.10) c d |x| 2−d ∫ |x| 2 0 s d 2 −2 e −s ds ≤ c d |x| 2−d ( ) d Γ − 1 ≤ (c 2 √ d/|x|) d−2 , using the asymptotic behavior of the Gamma function in the last step, cf. =-=[14]-=-, p. 88. As for the first integral in the last line of (1.9), we note that for 1 ≤ s ≤ κ|x|, the function s → −d |x| log s − |x| log(1 + κ ) has derivative 2 s − d |x| + 2s s κ|x| s + κ|x| s≤κ|x| ≥ − ... |

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Citation Context ...bound, (with m = |y|1 ≤ d 2 ): (3.19) Py[H S1(0,d 2 ) < ˜ HB1(0,|y|1)] ≥ Q|y|1[H d 2 < ˜ H|y|1] = pm(1 + ρm+1 + ρm+1 ρm+2 + · · · + ρm+1 . . .ρ d 2 −1) −1 , where ρℓ = qℓ, for ℓ ≥ 0, and we have used =-=[5]-=-, (5), p. 73. pℓ Note that the expression in the right-hand side of (3.19) is a decreasing function of each ρℓ, m + 1 < ℓ < d2 . If we further observe that ρℓ ≤ ( 1 1 1 − × 2 2 4 )(1 1 1 + × 2 2 4 )−1... |

157 | Intersections of random walks
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Citation Context ...nction of random interlacements at a level u0 close to log d, see the proof of Theorem 3.1. 4) The asymptotic behavior of g(x) for d fixed and large x is well-known, see for instance [10], p. 313, or =-=[12]-=-, p. 31: lim x→∞ g(x) d = |x| d−2 2 Γ ( ) d d − − 1 π 2 . 2 The asymptotic behavior of g(·) at the origin, or close to the origin when d tends to infinity is also well-known, see for instance [13], p.... |

75 |
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Citation Context ...ower bound, and not an upper bound, because the density of V u decreases with u). Whereas the required lower bound in the Bernoulli percolation context follows from a short Peierls-type argument, cf. =-=[3]-=-, p. 640, [11], p. 222, or [8], p. 25, the proof of (0.6) for random interlacements is quite involved. The long range dependence present in the model is deeply felt. An important feature of working in... |

44 | Percolation on finite graphs and isoperimetric inequalities
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(Show Context)
Citation Context ...whether the following reinforcement of (0.4) actually holds: P[0 ∈ V u∗ ] = e −u∗/g(0) ∼ (2d) −1 , as d → ∞ . This would indicate a similar high-dimensional behavior as for Bernoulli percolation, see =-=[1]-=-, [2], [6], [9], [11]. In the case of interlacement percolation on a 2d-regular tree, such an asymptotic behavior is known to hold, cf. [22]. □ 24A Appendix In this appendix we prove an elementary in... |

32 |
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Citation Context ...e connectivity function of random interlacements at a level u0 close to log d, see the proof of Theorem 3.1. 4) The asymptotic behavior of g(x) for d fixed and large x is well-known, see for instance =-=[10]-=-, p. 313, or [12], p. 31: lim x→∞ g(x) d = |x| d−2 2 Γ ( ) d d − − 1 π 2 . 2 The asymptotic behavior of g(·) at the origin, or close to the origin when d tends to infinity is also well-known, see for ... |

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Citation Context ...f Proposition 2.3. The bounds on the Harnack constants are derived in Proposition 1.3 with the help of the general Lemma A.2 from the Appendix, which is an adaptation of Lemma 10.2 in Grigoryan-Telcs =-=[7]-=-. Let us now describe how this article is organized. In Section 1 we introduce notation and recall several useful facts concerning random walks and random interlacements. An important role is played b... |

24 | Vacant set of random interlacements and percolation
- Sznitman
(Show Context)
Citation Context ...d P[AT 2 (µ2,2)] = P[AT 2 (µ′ 2,2 + µ∗2,2 )], which appears in the right-hand side of (2.22), in terms of pn(u ′) when u − u ′ is not too small. For this purpose we employ the sprinkling technique of =-=[18]-=-, and loosely speaking establish that µ ∗ 2,2 dominates “up to small corrections” the contribution of µ′ 2,1 + µ′ 1,2 in P[Au′ ] = T 2 P[AT 2 (µ′ 2,2 + µ′ 2,1 + µ′ 1,2)]. With this in mind we define a... |

17 | Upper bound on the disconnection time of discrete cylinders and random interlacements, Annals of Probability, to appear, preprint available at http://www.math.ethz.ch/u/sznitman/preprints
- Sznitman
(Show Context)
Citation Context ...anuary 2010 ETH Zürich CH-8092 Zürich Switzerlands0 Introduction Random interlacements have proven useful in understanding how trajectories of random walks can create large separating interfaces, see =-=[19]-=-, [20], [4]. In the case of Z d , d ≥ 3, it is known that the interlacement at level u ≥ 0, is a random subset of Z d , which is connected, ergodic under translations, and infinite when u is positive,... |

14 | Percolation for the vacant set of random interlacements
- Sidoravicius, Sznitman
- 2009
(Show Context)
Citation Context ...MENT PERCOLATION IN HIGH DIMENSION Alain-Sol Sznitman Preliminary Draft Abstract The vacant set of random interlacements on Z d , d ≥ 3, has non-trivial percolative properties. It is known from [18], =-=[16]-=-, that there is a non-degenerate critical value u∗ such that the vacant set at level u percolates when u < u∗ and does not percolate when u > u∗. We derive here an asymptotic upper bound on u∗, as d g... |

12 |
Random walks in multidimensional spaces, especially on periodic lattices
- Montroll
- 1956
(Show Context)
Citation Context ... or [12], p. 31: lim x→∞ g(x) d = |x| d−2 2 Γ ( ) d d − − 1 π 2 . 2 The asymptotic behavior of g(·) at the origin, or close to the origin when d tends to infinity is also well-known, see for instance =-=[13]-=-, p. 246, or [21], Remark 1.3 1). On the other hand the behavior of g(·) at intermediate scales when d tends to infinity seems much less explored. □ The bounds on the Green function of Lemma 1.1 toget... |

12 | On the uniqueness of the infinite cluster of the vacant set of random interlacements
- Teixeira
(Show Context)
Citation Context ... high-dimensional behavior as for Bernoulli percolation, see [1], [2], [6], [9], [11]. In the case of interlacement percolation on a 2d-regular tree, such an asymptotic behavior is known to hold, cf. =-=[22]-=-. □ 24A Appendix In this appendix we prove an elementary inequality, which enters the proof of the Green function estimate (1.14), see Lemma A.1 below. We then prove in Lemma A.2 a bound on Harnack c... |

12 | Interlacement percolation on transient weighted graphs, preprint available at http://www.math.ethz.ch/∼teixeira
- Teixeira
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Citation Context ...valent to log d for large d, and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees, which has been explicitly computed in =-=[23]-=-. Departement Mathematik January 2010 ETH Zürich CH-8092 Zürich Switzerlands0 Introduction Random interlacements have proven useful in understanding how trajectories of random walks can create large s... |

9 |
Heat kernels of graphs
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- 1993
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Citation Context ...transition probabilities. Relating the continuous and the discrete time random walks on Z d , we thus find that ∫ ∞ (1.7) g(x) = d 0 d∏ pt(0, xi)dt, for x = (x1, . . ., xd) ∈ Zd . From Theorem 3.5 of =-=[15]-=-, and the fact that the function i=1 F(γ) = − log ( γ + √ γ 2 + 1 ) + 1 γ (√ γ 2 + 1 − 1 ) , γ > 0 , that appears in Theorem 3.5 of [15], has derivative −(1+ √ γ2 + 1) −1 , tends to 0 in γ = 0, and th... |

7 |
Percolation in high dimensions
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(Show Context)
Citation Context ...er the following reinforcement of (0.4) actually holds: P[0 ∈ V u∗ ] = e −u∗/g(0) ∼ (2d) −1 , as d → ∞ . This would indicate a similar high-dimensional behavior as for Bernoulli percolation, see [1], =-=[2]-=-, [6], [9], [11]. In the case of interlacement percolation on a 2d-regular tree, such an asymptotic behavior is known to hold, cf. [22]. □ 24A Appendix In this appendix we prove an elementary inequal... |

7 | Percolation in high dimensions
- Gordon
- 1991
(Show Context)
Citation Context ...e following reinforcement of (0.4) actually holds: P[0 ∈ V u∗ ] = e −u∗/g(0) ∼ (2d) −1 , as d → ∞ . This would indicate a similar high-dimensional behavior as for Bernoulli percolation, see [1], [2], =-=[6]-=-, [9], [11]. In the case of interlacement percolation on a 2d-regular tree, such an asymptotic behavior is known to hold, cf. [22]. □ 24A Appendix In this appendix we prove an elementary inequality, ... |

7 | On the domination of random walk on a discrete cylinder by random interlacements
- Sznitman
(Show Context)
Citation Context ... 2010 ETH Zürich CH-8092 Zürich Switzerlands0 Introduction Random interlacements have proven useful in understanding how trajectories of random walks can create large separating interfaces, see [19], =-=[20]-=-, [4]. In the case of Z d , d ≥ 3, it is known that the interlacement at level u ≥ 0, is a random subset of Z d , which is connected, ergodic under translations, and infinite when u is positive, see [... |

6 |
Asymptotics in high dimension for percolation, in Disorder in Physical Systems
- Kesten
- 1990
(Show Context)
Citation Context ...nd not an upper bound, because the density of V u decreases with u). Whereas the required lower bound in the Bernoulli percolation context follows from a short Peierls-type argument, cf. [3], p. 640, =-=[11]-=-, p. 222, or [8], p. 25, the proof of (0.6) for random interlacements is quite involved. The long range dependence present in the model is deeply felt. An important feature of working in high dimensio... |

6 | Connectivity bounds for the vacant set of random interlacements
- Sidoravicius, Sznitman
- 2010
(Show Context)
Citation Context ...far away) x are linked by a path in V u , (i.e. the probability of a vacant crossing at level u between 0 and x), has a stretched exponential decay in x, when u is bigger than u∗∗, see Theorem 0.1 of =-=[17]-=-. We will briefly comment on the proof of Theorem 0.1. In view of (0.3), we only need to show that (0.6) lim sup d u∗/ log d ≤ 1 . As for Bernoulli bond or site percolation, similarities between what ... |

5 |
Mean-field critical phenomena for percolation in high dimensions
- Hara, Slade
- 1990
(Show Context)
Citation Context ...lowing reinforcement of (0.4) actually holds: P[0 ∈ V u∗ ] = e −u∗/g(0) ∼ (2d) −1 , as d → ∞ . This would indicate a similar high-dimensional behavior as for Bernoulli percolation, see [1], [2], [6], =-=[9]-=-, [11]. In the case of interlacement percolation on a 2d-regular tree, such an asymptotic behavior is known to hold, cf. [22]. □ 24A Appendix In this appendix we prove an elementary inequality, which... |

4 |
Giant vacant component left by a random walk in a random d-regular graph
- Čern´y, Teixeira, et al.
(Show Context)
Citation Context ...ETH Zürich CH-8092 Zürich Switzerlands0 Introduction Random interlacements have proven useful in understanding how trajectories of random walks can create large separating interfaces, see [19], [20], =-=[4]-=-. In the case of Z d , d ≥ 3, it is known that the interlacement at level u ≥ 0, is a random subset of Z d , which is connected, ergodic under translations, and infinite when u is positive, see [18]. ... |

3 | A lower bound on the critical parameter of interlacement percolation in high dimension
- Sznitman
- 2011
(Show Context)
Citation Context ...that the vacant set at level u percolates when u < u∗ and does not percolate when u > u∗. We derive here an asymptotic upper bound on u∗, as d goes to infinity, which complements the lower bound from =-=[21]-=-. Our main result shows that u∗ is equivalent to log d for large d, and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on 2d-regular trees,... |

3 |
Connectivity boundsforthevacantset ofrandom interlacements
- Sznitman
(Show Context)
Citation Context ...far away) x are linked by a path in V u , (i.e. the probability of a vacant crossing at level u between 0 and x), has a stretched exponential decay in x, when u is bigger than u∗∗, see Theorem 0.1 of =-=[17]-=-. We will briefly comment on the proof of Theorem 0.1. In view of (0.3), we only need to show that (0.6) limsup d u∗/logd ≤ 1. As for Bernoulli bond or site percolation, similarities between what happ... |

2 |
Telcs. Sub-Gaussianestimates of heat kernels oninfinite graphs
- GrigoryanandA
(Show Context)
Citation Context ...f Proposition 2.3. The bounds on the Harnack constants are derived in Proposition 1.3 with the help of the general Lemma A.2 from the Appendix, which is an adaptation of Lemma 10.2 in Grigoryan-Telcs =-=[7]-=-. Let us now describe how this article is organized. In Section 1 we introduce notation and recall several useful facts concerning random walks and random interlacements. An important role is played b... |