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2019 Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup
Caio Alves, Artem Sapozhnikov
Electron. J. Probab. 24: 1-34 (2019). DOI: 10.1214/19-EJP360

Abstract

It has been recently understood [9, 24, 30] that for a general class of percolation models on $\mathbb{Z} ^{d}$ 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 $\mathbb{Z} ^{d}$ in dimensions $d\geq 3$. 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 the required decoupling inequalities are not valid.

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.

Citation

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Caio Alves. Artem Sapozhnikov. "Decoupling inequalities and supercritical percolation for the vacant set of random walk loop soup." Electron. J. Probab. 24 1 - 34, 2019. https://doi.org/10.1214/19-EJP360

Information

Received: 13 August 2018; Accepted: 8 September 2019; Published: 2019
First available in Project Euclid: 3 October 2019

zbMATH: 07142904
MathSciNet: MR4017128
Digital Object Identifier: 10.1214/19-EJP360

Subjects:
Primary: 60K35
Secondary: 60G55, 60J10, 82B43

JOURNAL ARTICLE
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