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.

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