  In this paper, we relate the coupling of Markov chains, at the basis of
perfect sampling methods, with damage spreading, which captures the chaotic
nature of stochastic dynamics. For two-dimensional spin glasses and hard
spheres we point out that the obstacle to the application of perfect-sampling
schemes is posed by damage spreading rather than by the survey problem of the
entire configuration space. We find dynamical damage-spreading transitions
deeply inside the paramagnetic and liquid phases, and show that critical values
of the transition temperatures and densities depend on the coupling scheme. We
discuss our findings in the light of a classic proof that for arbitrary Monte
Carlo algorithms damage spreading can be avoided through non-Markovian coupling
schemes.
