  We discuss convergence and coupling of Markov chains, and present general
relations between the transfer matrices describing these two processes. We then
analyze a recently developed local-patch algorithm, which computes rigorous
upper bound for the coupling time of a Markov chain for non-trivial
statistical-mechanics models. Using the coupling from the past protocol, this
allows one to exactly sample the underlying equilibrium distribution. For spin
glasses in two and three spatial dimensions, the local-patch algorithm works at
lower temperatures than previous exact-sampling methods. We discuss variants of
the algorithm which might allow one to reach, in three dimensions, the
spin-glass transition temperature. The algorithm can be adapted to hard-sphere
models. For two-dimensional hard disks, the algorithm allows us to draw exact
samples at higher densities than previously possible.
