Charting the replica symmetric phase

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.

We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].

This is a joint work with: A. Coja-Oghlan, N. Jaafari, M. Kang and T.Kapetanopoulos