- Start with a bond-percolation process just above criticality, say $p=1/2+\varepsilon$ on the graph $\mathbb Z^2$ with $\varepsilon>0$.
- Sample $D\in\{0,1\}^E$ from an independent product measure with $\mathbb P(D_e=1)=\delta>0 \;\forall e\in E$.
- Remove all edges $e$ with $D_e=1$ from the percolation process.
Clearly one ends up with another percolation process with parameter $p'=(1/2+\varepsilon)(1-\delta)<1/2$, i.e. below criticality (for $\varepsilon$ sufficiently small). In particular the resulting process has no infinite connected components almost surely.
Now what happens if the process $D$ of edges to be removed is not a product measure (but still essentially translation invariant and such that a positive fraction of edges are deleted)? Can we still choose $\varepsilon$ sufficiently small so that there won't be any infinite connected components after deleting these edges from our original percolation process?
(Partial answer) In case the law of $D$ stochastically dominates some (non-degenerate) Bernoulli product measure, the same reasoning as before applies.
(The problematic case) What if the law of $D$ is strongly "repellant"? An extreme example would be a law of $D$ where deleting a pair of adjacent edges has probability $0$, i.e. $\mathbb P(D_e=D_{e'}=1)=0$, whenever $e,e'$ share a vertex.
Edit: $D$ should still be independent of the original percolation process.