  We study the scaling properties of the non-equilibrium stationary states
(NESS) of a reaction-diffusion model. Under a suitable smallness condition, we
show that the density of particles satisfies a law of large numbers with
respect to the NESS, with an explicit rate of convergence, and we also show
that at mesoscopic scales the NESS is well approximated by a local equilibrium
(product) measure, in the total variation distance. In addition, in dimensions
$d \leq3$ we show a central limit theorem (CLT) for the density of particles
under the NESS. The corresponding Gaussian limit can be represented as an
independent sum of a white noise and a massive Gaussian free field, and in
particular it presents macroscopic correlations.
