No infinite clusters at criticality for two-dimensional dependent percolation under abstract hypotheses

We establish a general criterion ensuring the absence of infinite connected components at criticality for a broad class of two-dimensional dependent percolation models.

Our assumptions include monotonicity, stationarity and ergodicity, positive association, a quantitative decoupling condition, continuity in the percolation parameter, and trivial behavior at the extremes.

As application, we show how our criterion recovers known results for classical dependent percolation models, such as Gaussian level-set percolation and Boolean percolation.