Communications in Mathematical Physics

, Volume 352, Issue 3, pp 979–1017 | Cite as

Low Temperature Asymptotics of Spherical Mean Field Spin Glasses

  • Aukosh Jagannath
  • Ian TobascoEmail author


In this paper, we study the low temperature limit of the spherical Crisanti–Sommers variational problem. We identify the \({\Gamma}\)-limit of the Crisanti–Sommers functionals, thereby establishing a rigorous variational problem for the ground state energy of spherical mixed p-spin glasses. As an application, we compute moderate deviations of the corresponding minimizers in the low temperature limit. In particular, for a large class of models this yields moderate deviations for the overlap distribution as well as providing sharp interpolation estimates between models. We then analyze the ground state energy problem. We show that this variational problem is dual to an obstacle-type problem. This duality is at the heart of our analysis. We present the regularity theory of the optimizers of the primal and dual problems. This culminates in a simple method for constructing a finite dimensional space in which these optimizers live for any model. As a consequence of these results, we unify independent predictions of Crisanti–Leuzzi and Auffinger–Ben Arous regarding the one-step Replica Symmetry Breaking (1RSB) phase in this limit. We find that the “positive replicon eigenvalue” and “pure-like” conditions are together necessary for optimality, but that neither are themselves sufficient, answering a question of Auffinger and Ben Arous in the negative. We end by proving that these conditions completely characterize the 1RSB phase in 2 + p-spin models.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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