  On the level of an effective quark theory, we define confinement by the
absence of quark anti-quark thresholds in correlation functions. We then
propose a confining Nambu-Jona-Lasinio-type model. Its four fermion interaction
in the color channel is mediated by a random color matrix. We investigate the
model's phase structure as well as its behavior under extreme conditions, i.e.
high temperature and/or high density. In order to study the interplay of the
proposed mechanism of confinement and asymptotic freedom, we consider an
extended Gross-Neveu model. Quark anti-quark thresholds are proven to be absent
in the scalar correlation function implying that the decay of the scalar meson
into free quarks is avoided. The transition from the low energy to the high
energy perturbative regime is found to be smooth. Perturbation theory provides
a good description of Green's functions at high energies, although there is no
quark liberation.
