Replica Symmetry Breaking in Multi-species Sherrington–Kirkpatrick Model

  • Erik BatesEmail author
  • Leila Sloman
  • Youngtak Sohn


In the Sherrington–Kirkpatrick (SK) and related mixed p-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first non-asymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.


Spin glasses Sherrington–Kirkpatrick model de Almeida–Thouless line 

Mathematics Subject Classification

60K35 82B26 82B44 



We are grateful to Amir Dembo and Andrea Montanari for their advice and encouragement on this project. We thank Antonio Auffinger, Erwin Bolthausen, and Aukosh Jagannath for their insights and feedback, and the referee for several suggestions to improve the manuscript.


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Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of StatisticsStanford UniversityStanfordUSA

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