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Communications in Mathematical Physics

, Volume 359, Issue 2, pp 603–698 | Cite as

Charting the Replica Symmetric Phase

  • Amin Coja-Oghlan
  • Charilaos Efthymiou
  • Nor Jaafari
  • Mihyun Kang
  • Tobias Kapetanopoulos
Article

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. in Proc Natl Acad Sci 104:10318–10323, 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. in Phys Rev E 84:066106, 2011).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  • Charilaos Efthymiou
    • 1
  • Nor Jaafari
    • 1
  • Mihyun Kang
    • 2
  • Tobias Kapetanopoulos
    • 1
  1. 1.Mathematics InstituteGoethe UniversityFrankfurtGermany
  2. 2.Institute of Discrete MathematicsTechnische Universität GrazGrazAustria

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