The ABC model is a one-dimensional lattice gas, where each site is occupied by one of three types of species (A, B or C). The dynamics is fully specified by exchange rates between neighboring sites. The number of each species is conserved, hence properties of the system depend on three global densities. The model is considered on a ring of \(L\) sites with periodic boundary conditions.The system shows long range correlations and exhibits a second-order phase transition. The goal of the present work was to analyze these long range correlations at and in the neighborhood of a phase transition, when their decay is no longer characterized by a standard \(1/L\) scale and shows an anomalous size-dependence (non-integer inverse power of \(L\)). The main result is that the pertinent anomalous scaling can be understood by an effective theory in the authors’ article [“Current fluctuations at a phase transition”, Europhys. Lett. 96, No. 2, 20001 (2011; doi:10.1209/0295-5075/96/20001)] for the amplitude of the slow density mode which becomes unstable at the transition. Cumulants of the particle current are computed away from phase transition and shown to become singular as the transition is approached. This matches with already known features of the critical regime.