  We revisit here the naturalness problem of Lorentz invariance violations on a
simple toy model of a scalar field coupled to a fermion field via a Yukawa
interaction. We first review some well-known results concerning the low-energy
percolation of Lorentz violation from high energies, presenting some details of
the analysis not explicitly discussed in the literature and discussing some
previously unnoticed subtleties. We then show how a separation between the
scale of validity of the effective field theory and that one of Lorentz
invariance violations can hinder this low-energy percolation. While such
protection mechanism was previously considered in the literature, we provide
here a simple illustration of how it works and of its general features.
Finally, we consider a case in which dissipation is present, showing that the
dissipative behaviour does not percolate generically to lower mass dimension
operators albeit dispersion does. Moreover, we show that a scale separation can
protect from unsuppressed low-energy percolation also in this case.
