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 (Disorder chaos and multiple valleys in spin glasses, 2009) and Ding et al. (Ann Probab 43(6):3468–3493, 2015), 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.
Bibliographical noteFunding Information:
The authors are indebted to N. Krylov and M. Safonov for illuminating discussions on the regularity properties of the Parisi PDE. They thank S. Chatterjee, D. Panchenko, and the anonymous referees for a number of suggestions regarding the presentation of the paper. The research of W.-K. C. is partially supported by NSF Grant DMS-16-42207 and Hong Kong Research Grants Council GRF-14-302515. The research of M. H. and G. L. is partially supported by NSF Grant DMS-14-18386.
© 2017, Springer-Verlag Berlin Heidelberg.
- Disorder chaos
- Energy landscape
- Multiple peaks
- Parisi formula
- Sherrington–Kirkpatrick model