Conditional decoupling of random interlacements

Alves, Caio; Popov, Serguei

doi:10.30757/alea.v15-38

Cited by 6 publications

(11 citation statements)

References 21 publications

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“…They provide a connection between the measures P u with different values of the parameter and serve only to prove the likeliness of certain patterns in S ∞ , cf. [27, Remark 1.9 (1)]. More precisely, if an increasing, resp.…”

Section: Proof Of Lemma 56mentioning

confidence: 99%

“…The fundamental idea behind the major progress in understanding these models (which are monotone in their intensity parameters) is that the effect of correlations can be well dominated with a slight tilt of the intensity parameter (sprinkling). This idea is formalized in correlation inequalities, known as decoupling inequalities [31,32,26,10,21,22,1]. A general class of percolation models, which satisfy a suitable decoupling inequality and contains the three models mentioned above, was considered in [8,23,27], where most of the geometric properties of the infinite percolation cluster, previously only known to hold for Bernoulli percolation, were proven.…”

mentioning

confidence: 99%

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Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

Alves¹,

Sapozhnikov²

2019

Electron. J. Probab.

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It has been recently understood [8,23,27] that for a general class of percolation models on Z d satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT.In this paper we consider the random walk loop soup on Z d in dimensions d ≥ 3. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since it does not satisfy the required decoupling inequalities.We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.)Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

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Section: Proof Of Lemma 56mentioning

confidence: 99%

mentioning

confidence: 99%

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

Alves¹,

Sapozhnikov²

2019

Electron. J. Probab.

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“…k records the successive steps of the SRW S in the ith direction. Obviously, we have that S (1) , S (2) and S (3) are independent (the same is not true for S(i) ).…”

Section: Lemma 24mentioning

confidence: 97%

“…and we define T (x, M, n) = T (1) (x, M, n) ∩ T (2) (x, M, n) ∩ T (3) (x, M, n). Not focusing, for now, on technicalities, the key part of the event T (x, M, n) is that, not far away from x in the direction of the drift, there is a structure in I u creating a dead end for the biased random walk.…”

Section: Trapsmentioning

confidence: 99%

“…That is, we need to be able to work with the conditional law of the interlacement set, given that inside some finite set the interlacement configuration is (partially or even completely) revealed. Next, we formulate a result from [1] about the conditional decoupling for random interlacements. We also observe that the unconditional decoupling from [13] is not enough in this situation.…”

Section: Finding Traps In the Interlacement Setmentioning

confidence: 99%

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Biased random walks on the interlacement set

Fribergh¹,

Popov²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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We study a biased random walk on the interlacement set of Z d for d ≥ 3. Although the walk is always transient, we can show, in the case d = 3, that for any value of the bias the walk has a zero limiting speed and actually moves slower than any power.Résumé. Nous étudions la marche biaisée sur un entrelac aléatoire de Z d avec d ≥ 3. Nous montrons que la marche est transiente mais que, dans le cas d = 3, elle est sous-ballistique pour toutes les valeurs du biais et que ses déplacements sont inférieurs à n'importe quel polynôme.

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An Improved Decoupling Inequality for Random Interlacements

2019

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In this paper we obtain a decoupling feature of the random interlacements process I u ⊂ Z d , at level u, d ≥ 3. More precisely, we show that the trace of the random interlacements process on two disjoint finite sets, F and its translated F + x, can be coupled with high probability of success, when x is large, with the trace of a process of independent excursions, which we call the noodle soup process. As a consequence, we obtain an upper bound on the covariance between two [0, 1]-valued functions depending on the configuration of the random interlacements on F and F + x, respectively. This improves a previous bound obtained by Sznitman in [9].

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Conditional decoupling of random interlacements

Cited by 6 publications

References 21 publications

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup

Biased random walks on the interlacement set

An Improved Decoupling Inequality for Random Interlacements

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