Erratum: The random field XY model on sparse random graphs shows replica symmetry breaking and marginally stable ferromagnetism (Journal of Physics A: Mathematical and Theoretical (2019) 52 (284001)

After the publication, the authors have discovered an error made in the analysis of the large C limit: namely equation (A5) in the appendix was solved only perturbatively around the trivial solution. In fact, a non-trivial solution can also appear discontinuously at low enough temperatures, corresponding to a first-order thermodynamical transition. The discovery of such a first-order transition makes necessary to reformulate the paragraph where the data in figure 4 are discussed and to amend the appendix with the right computation and the appropriate phase diagram. The last paragraph of section 3.5 after the sentence ‘the non-monotonicity of the critical line between the paramagnetic and the ferromagnetic line’ should read: ‘As discussed in detail in appendix A, the curve we are drawing in figure 4 is actually the spinodal curve of the paramagnetic phase. In the large C limit it becomes non-monotonic as a consequence of the phase transition becoming first order at low enough temperatures. In appendix A we show the exact location of the coexistence phase in the C = 8limit. For finite C values we have just checked that the saddle point equations admit a unique solution, i.e. (Figure Presented) (Formula Presented).