  We examine the behavior of the 2-spin spherical Sherrington-Kirkpatrick model
with an external field by analyzing the overlap of a spin with the external
field. Previous research has noted that, at low temperature, this overlap
exhibits dramatically different behavior in the presence of an external field
as compared to the model with no external field. The transition between those
two settings was examined in a recent physics paper by Baik, Collins-Woodfin,
Le Doussal, and Wu as well as a recent math paper by Landon and Sosoe. Both
papers focus on the setting in which the external field strength, $h$,
approaches zero as the dimension, $N$, approaches infinity. In particular, the
paper of Baik et al studies the overlap with a microscopic external field
($h\sim N^{-1/2}$) but without a rigorous proof. This paper aims to give a
proof of that result. The proof involves representing the generating function
of the overlap as a ratio of contour integrals and then analyzing the
asymptotics of those contour integrals using results from random matrix theory.
