  We consider a quantum system in contact with a heat bath consisting in an
infinite chain of identical sub-systems at thermal equilibrium at inverse
temperature $\beta$. The time evolution is discrete and such that over each
time step of duration $\tau$, the reference system is coupled to one new
element of the chain only, by means of an interaction of strength $\lambda$. We
consider three asymptotic regimes of the parameters $\lambda$ and $\tau$ for
which the effective evolution of observables on the small system becomes
continuous over suitable macroscopic time scales $T$ and whose generator can be
computed: the weak coupling limit regime $\lambda\ra 0$, $\tau=1$, the regime
$\tau\ra 0$, $\lambda^2\tau \ra 0$ and the critical case $\lambda^2\tau=1$,
$\tau\ra 0$. The first two regimes are perturbative in nature and the effective
generators they determine is such that a non-trivial invariant sub-algebra of
observables naturally emerges. The third asymptotic regime goes beyond the
perturbative regime and provides an effective dynamics governed by a general
Lindblad generator naturally constructed from the interaction Hamiltonian.
Conversely, this result shows that one can attach to any Lindblad generator a
repeated quantum interactions model whose asymptotic effective evolution is
generated by this Lindblad operator.
