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On coupling and “Vacant set level set” percolation. (English) Zbl 1412.60135
Summary: 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 $${\mathbb Z}^d$$, $$d \ge 3$$.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G15 Gaussian processes 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 82B43 Percolation
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