Free Energy and Overlaps of a Spherical Spin Glass with an External Field

Abstract

In this thesis, we analyze the free energy and the overlaps in the 2-spin spherical Sherrington Kirkpatrick (SSK) 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, h, approaches zero as the dimension, N, approaches infinity. In particular, we consider overlaps with the external field, the ground state, and a replica. The 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 and we discuss the implications of the results for susceptibility and for the geometry of the Gibbs measure.

The analysis throughout this thesis focuses on the SSK model with external field strength proportional to N^(-a) for 0< a<1. We find that the free energy exhibits a transition at a=1/6 in the low temperature case but at a=1/4 at the high temperature case. Furthermore, the overlaps do not exhibit any transition in the high temperature case, but exhibit two transitions in the low temperature case, at a=1/6 and a=1/2. These scalings are referred to as the mesoscopic and microscopic external field respectively. In the final chapter, we present a more detailed analysis of the overlaps in the microscopic setting.