  We study a spin system with both mixed even-spin Sherrington-Kirkpatrick (SK)
couplings and Curie-Weiss (CW) interaction. Our main results are: (i) The
thermodynamic limit of the free energy is given by a variational formula
involving the free energy of the SK model with a change in the external field.
(ii) In the presence of a centered Gaussian external field, the positivity of
the overlap and the extended Ghirlanda-Guerra identities hold on a dense subset
of the temperature parameters. (iii) We establish a general inequality between
the magnetization and overlap. (iv) We construct a temperature region in which
the magnetization can be quantitatively controlled and deduce different senses
of convergence for the magnetization depending on whether the external field is
present or not. Our approach is based on techniques from the study of the CW
and SK models and results in convex analysis.
