In 1979, G. Parisi predicted a variational formula for the thermodynamic limit of the free energy in the Sherrington-Kirkpatrick model and described the role played by its minimizer. This formula was verified in the seminal work of Talagrand and later generalized to the mixed p-spin models by Panchenko. In this paper, we prove that the minimizer in Parisi's formula is unique at any temperature and external field by establishing the strict convexity of the Parisi functional.
We investigate the energy landscape of the mixed even p-spin model with Ising spin configurations.
We show that for any given energy level between zero and the maximal energy, with
overwhelming probability there exist exponentially many distinct spin configurations such that
their energies stay near this energy level. Furthermore, their magnetizations and overlaps are
concentrated around some fixed constants. In particular, at the level of maximal energy, we
prove that the Hamiltonian exhibits exponentially many orthogonal peaks. This improves the
results of Chatterjee  and Ding-Eldan-Zhai , where the former established a logarithmic
size of the number of the orthogonal peaks, while the latter proved a polynomial size. Our second
main result obtains disorder chaos at zero temperature and at any external field. As a byproduct,
this implies that the fluctuation of the maximal energy is superconcentrated when the external
field vanishes and obeys a Gaussian limit law when the external field is present.