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Anomalous long-range correlations at a non-equilibrium phase transition

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.