  We discuss coupling from the past techniques (CFTP) for perturbations of
interacting particle systems on the d-dimensional integer lattice, with a
finite set of states, within the framework of the graphical construction of the
dynamics based on Poisson processes. We first develop general results for what
we call CFTP times with ambiguities. These are analogous to classical coupling
(from the past) times, except that the coupling property holds only provided
that some ambiguities concerning the stochastic evolution of the system are
resolved. If these ambiguities are rare enough on average, CFTP times with
ambiguities can be used to build actual CFTP times, whose properties can be
controlled in terms of those of the original CFTP time with ambiguities. We
then prove a general perturbation result, which can be stated informally as
follows. Start with an interacting particle system possessing a CFTP time whose
definition involves the exploration of an exponentially integrable number of
points in the graphical construction, and which satisfies the positive rates
property. Then consider a perturbation obtained by adding new transitions to
the original dynamics. Our result states that, provided that the perturbation
is small enough (in the sense of small enough rates), the perturbed interacting
particle system too possesses a CFTP time (with nice properties such as an
exponentially decaying tail). The proof consists in defining a CFTP time with
ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed
dynamics. Finally, we discuss examples of particle systems to which this result
can be applied. Concrete examples include a class of neighbor-dependent
nucleotide substitution model, and variations of the classical voter model,
illustrating the ability of our approach to go beyond the case of weakly
interacting particle systems.
