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.
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.
Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for ran- dom walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance prin- ciples for domains with more general boundaries.
Ruihan GuoSchool of Mathematics and Statistics, Zhengzhou UniversityYan XuSchool of Mathematical Sciences, University of Science and Technology of ChinaZhengfu XuDepartment of Mathematical, Michigan Technological University