  We are interested in phase transitions in certain percolation models on point
processes and their dependence on clustering properties of the point processes.
We show that point processes with smaller void probabilities and factorial
moment measures than the stationary Poisson point process exhibit non-trivial
phase transition in the percolation of some coverage models based on level-sets
of additive functionals of the point process. Examples of such point processes
are determinantal point processes, some perturbed lattices, and more generally,
negatively associated point processes. Examples of such coverage models are
$k$-coverage in the Boolean model (coverage by at least $k$ grains) and
SINR-coverage (coverage if the signal-to-interference-and-noise ratio is
large). In particular, we answer in affirmative the hypothesis of existence of
phase transition in the percolation of $k$-faces in the \v{C}ech simplicial
complex (called also clique percolation) on point processes which cluster less
than the Poisson process. We also construct a Cox point process, which is "more
clustered" than the Poisson point process and whose Boolean model percolates
for arbitrarily small radius. This shows that clustering (at least, as detected
by our specific tools) does not always "worsen" percolation, as well as that
upper-bounding this clustering by a Poisson process is a consequential
assumption for the phase transition to hold.
