  We consider $Z^d$, with d bigger or equal to three. We investigate the vacant
set of random interlacements in the strongly percolative regime, the vacant set
of the simple random walk, and the excursion set above a given level of the
Gaussian free field in the strongly percolative regime. We derive asymptotic
upper and lower exponential bounds for the large deviation probability that the
adequately thickened component of the boundary of a large box centered at the
origin in the respective vacant sets or excursion set leaves in the box a
macroscopic volume in its complement. We also derive geometric information on
the shape of the left-out volume. It is plausible, but open at the moment, that
certain critical levels coincide, both in the case of random interlacements and
of the Gaussian free field. If this holds true, the asymptotic upper and lower
bounds that we obtain are matching in principal order for all three models, and
the macroscopic holes are nearly spherical. We heavily rely on the recent work
arXiv:1706.07229 by Maximilian Nitzschner and the author for the coarse
graining procedure, which we employ in the derivation of the upper bounds.
