  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.
