Low Temperature Asymptotics of Spherical Mean Field Spin Glasses
Abstract
In this paper, we study the low temperature limit of the spherical CrisantiSommers variational problem. We identify the {Γ}limit of the CrisantiSommers functionals, thereby establishing a rigorous variational problem for the ground state energy of spherical mixed pspin 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 obstacletype 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 CrisantiLeuzzi and AuffingerBen Arous regarding the onestep Replica Symmetry Breaking (1RSB) phase in this limit. We find that the "positive replicon eigenvalue" and "purelike" 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 + pspin models.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 June 2017
 DOI:
 10.1007/s0022001728643
 arXiv:
 arXiv:1602.00657
 Bibcode:
 2017CMaPh.352..979J
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 Commun. Math. Phys. (2017) 352: 979