Annales Henri Poincaré

, Volume 16, Issue 3, pp 691–708 | Cite as

Multi-Species Mean Field Spin Glasses. Rigorous Results

  • Adriano Barra
  • Pierluigi Contucci
  • Emanuele MingioneEmail author
  • Daniele Tantari


We study a multi-species spin glass system where the density of each species is kept fixed at increasing volumes. The model reduces to the Sherrington–Kirkpatrick one for the single species case. The existence of the thermodynamic limit is proved for all density values under a convexity condition on the interaction. The thermodynamic properties of the model are investigated and the annealed, the replica-symmetric and the replica symmetry breaking bounds are proved using Guerra’s scheme. The annealed approximation is proved to be exact under a high-temperature condition. We show that the replica-symmetric solution has negative entropy at low temperatures. We study the properties of a suitably defined replica symmetry breaking solution and we optimize it within a novel ziggurat ansatz. The generalized order parameter is described by a Parisi-like partial differential equation.


Thermodynamic Limit Spin Glass Annealed Approximation Replica Symmetry Spin Glass Model 
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Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Adriano Barra
    • 1
  • Pierluigi Contucci
    • 3
  • Emanuele Mingione
    • 3
    Email author
  • Daniele Tantari
    • 2
  1. 1.Dipartimento di FisicaSapienza Università di RomaRomeItaly
  2. 2.Dipartimento di MatematicaSapienza Università di RomaRomeItaly
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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