  We consider the locally thinned Bernoulli field on $\mathbb Z^d$, which is
the lattice version of the Type-I Mat\'ern hardcore process in Euclidean space.
It is given as the lattice field of occupation variables, obtained as image of
an i.i.d. Bernoulli lattice field with occupation probability $p$, under the
map which removes all particles with neighbors, while keeping the isolated
particles. We prove that the thinned measure has a Gibbsian representation and
provide control on its quasilocal dependence, both in the regime of small $p$,
but also in the regime of large $p$, where the thinning transformation changes
the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness
criteria, disagreement percolation arguments, and cluster expansions.
