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2019 On coupling and “vacant set level set” percolation
Alain-Sol Sznitman
Electron. Commun. Probab. 24: 1-12 (2019). DOI: 10.1214/19-ECP217


In this note we discuss “vacant set level set” percolation on a transient weighted graph. It interpolates between the percolation of the vacant set of random interlacements and the level set percolation of the Gaussian free field. We employ coupling and derive a stochastic domination from which we deduce in a rather general set-up a certain monotonicity property of the percolation function. In the case of regular trees this stochastic domination leads to a strict inequality between some eigenvalues related to Ornstein-Uhlenbeck semi-groups for which we have no direct analytical proof. It underpins a certain strict monotonicity property that has significant consequences for the percolation diagram. It is presently open whether a similar looking diagram holds in the case of ${\mathbb Z}^d$, $d \ge 3$.


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Alain-Sol Sznitman. "On coupling and “vacant set level set” percolation." Electron. Commun. Probab. 24 1 - 12, 2019.


Received: 24 August 2018; Accepted: 14 February 2019; Published: 2019
First available in Project Euclid: 2 April 2019

zbMATH: 1412.60135
MathSciNet: MR3940195
Digital Object Identifier: 10.1214/19-ECP217

Primary: 60G15, 60J10, 60K35, 82B43


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