Charting the replica symmetric phase

Understand

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph.

• Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science.
• In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007].
• In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the $k$-XORSAT model and the diluted $k$-spin model for even $k$.