  The long-time dynamics of reaction-diffusion processes in low dimensions is
dominated by fluctuation effects. The one-dimensional coagulation-diffusion
process describes the kinetics of particles which freely hop between the sites
of a chain and where upon encounter of two particles, one of them disappears
with probability one. The empty-interval method has, since a long time, been a
convenient tool for the exact calculation of time-dependent particle densities
in this model. We generalise the empty-interval method by considering the
probability distributions of two simultaneous empty intervals at a given
distance. While the equations of motion of these probabilities reduce for the
coagulation-diffusion process to a simple diffusion equation in the continuum
limit, consistency with the single-interval distribution introduces several
non-trivial boundary conditions which are solved for the first time for
arbitrary initial configurations. In this way, exact space-time-dependent
correlation functions can be directly obtained and their dynamic scaling
behaviour is analysed for large classes of initial conditions.
