Spherical spin glass model with external field

Abstract

We analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick spin glass model with an external field for the purpose of understanding the transition between this model and the one without an external field. We compute the limiting values and fluctuations of the free energy as well as three types of overlaps in the setting where the strength of the external field goes to zero as the dimension of the spin variable grows. In particular, we consider overlaps with the external field, the ground state, and a replica. Our methods involve a contour integral representation of the partition function along with random matrix techniques. We also provide computations for the matching between different scaling regimes. Finally, we discuss the implications of our results for susceptibility and for the geometry of the Gibbs measure. Some of the findings of this paper are confirmed rigorously by Landon and Sosoe in their recent paper which came out independently and simultaneously.

arXiv comment75 pages, 9 figures, 5 tables. Version 2: a few small typos are corrected. One author's last name (Collins-Woodfin) is changed from version 1. Version 3: corrected a few small typos and added one reference [11]

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Cited by in corpus (10)

• Equilibrium Fluctuations in Mean-field Disordered Models
• Rank one HCIZ at high temperature: interpolating between classical and free convolutions
• Free energy fluctuations of the $2$-spin spherical SK model at critical temperature
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass
• Infinite volume Gibbs states and metastates of the random field mean-field spherical model
• Overlap of a spherical spin glass model with microscopic external field
• Mean Field Spin Glass Models under Weak External Field
• Fluctuations of the free energy of the spherical Sherrington-Kirkpatrick model with heavy-tailed interaction
• Optimization landscape in the simplest constrained random least-square problem
• Generalised Gibbs Ensemble for spherically constrained harmonic models