We study the antiferromagnetic Potts model on the Poissonian Erdos-Rényi random graph. By identifying a suitable interpolation structure and an extended variational principle, together with a positive temperature second-moment analysis we prove the existence of a phase transition at a positive critical temperature. Upper and lower bounds on the temperature critical value are obtained from the stability analysis of the replica symmetric solution (recovered in the framework of Derrida-Ruelle probability cascades) and from an entropy positivity argument.
Contucci, P., Dommers, S., Giardinà, C., & Starr, S. (2013). Antiferromagnetic Potts model on the Erdős-Rényi random graph. Communications in Mathematical Physics, 323(2), 517-554. https://doi.org/10.1007/s00220-013-1778-y