Current fluctuations at a phase transition

The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling.