Title: Charting the replica symmetric phase,
Abstract: 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].