On coupling and vacant set level set percolation

TL;DR

This paper investigates vacant set level set percolation on weighted graphs, exploring its relation to random interlacements and Gaussian free fields, and establishing monotonicity properties with implications for percolation theory.

Contribution

It introduces a coupling approach and stochastic domination to analyze percolation functions, revealing new monotonicity properties and eigenvalue inequalities, especially on regular trees.

Findings

Established a monotonicity property of the percolation function.

Derived eigenvalue inequalities related to Ornstein-Uhlenbeck semi-groups.

Identified open problems for percolation diagrams in higher dimensions.

Abstract

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 Z^d, with d bigger or equal to 3.