  Non-equilibrium diffusive systems are known to exhibit long-range
correlations, which decay like the inverse 1/L of the system size L in one
dimension. Here, taking the example of the ABC model, we show that this size
dependence becomes anomalous (the decay becomes a non-integer power of L) when
the diffusive system approaches a second-order phase transition.
  This power-law decay as well as the L-dependence of the time-time
correlations can be understood in terms of the dynamics of the amplitude of the
first Fourier mode of the particle densities. This amplitude evolves according
to a Langevin equation in a quartic potential, which was introduced in a
previous work to explain the anomalous behavior of the cumulants of the current
near this second-order phase transition. Here we also compute some of the
cumulants away from the transition and show that they become singular as the
transition is approached.
