Antiferromagnetic Potts model on the Erdos-Renyi random graph
  We study the antiferromagnetic Potts model on the Poissonian Erd\"os-R\'enyi 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 a positive entropy argument. 
Pierluigi Contucci, Sander Dommers, Cristian Giardina' and Shannon   Starr
math.PR cond-mat.dis-nn cond-mat.stat-mech math-ph math.MP