  We prove universality of the Ghirlanda-Guerra identities and spin
distributions in the mixed $p$-spin models. The assumption for the universality
of the identities requires exactly that the coupling constants have zero means
and finite variances, and the result applies to the Sherrington-Kirkpatrick
model. As an application, we obtain weakly convergent universality of spin
distributions in the generic $p$-spin models under the condition of two
matching moments. In particular, certain identities for 3-overlaps and
4-overlaps under the Gaussian disorder follow. Under the stronger mode of total
variation convergence, we find that universality of spin distributions in the
mixed $p$-spin models holds if mild dilution of connectivity by the Viana-Bray
diluted spin glass Hamiltonians is present and the first three moments of
coupling constants in the mixed $p$-spin Hamiltonians match. These universality
results are in stark contrast to the characterization of spin distributions in
the undiluted mixed $p$-spin models, which is known up to now that four
matching moments are required in general.
