Consider subcritical $(p > p_c)$ site percolation on $\mathbb{Z}^2$. Destroy the infinite open cluster, that is, make all its vertices closed. Then open the closed vertices independently from each other with probability delta. Let $\theta(p,\delta)$ denote the probability that the origin is in an infinite cluster in the configuration thus obtained.

We show that there is a positive delta such that $\theta(p, \delta) = 0$ for all $p > p_c$.

We also sketch how the methods in the proof of the result above can be used to deduce large deviation results on the volume large critical percolation clusters.

This is a joint work with Ioan Manolescu and Vladas Sidoravicius.