Some percolations involving the Gaussian free fields

Abstract

Consider an infinite, connected, locally finite graph with vertex set $\mathcal{V}$. Intuitively a simple point process on $\mathcal{V}$ with attractive properties, should percolate more easily than a Bernoulli point process with the same marginales. Although it seems wrong to imagine that it could be true in general, we confirm this intuition on several examples involving Gaussian free fields and permanental free fields.