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 [20] and Ding-Eldan-Zhai [29], 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.

Consider a spiked random tensor obtained as a mixture of two components: noise in the form of a symmetric Gaussian p-tensor and signal in the form of a symmetric low-rank random tensor. The latter low-rank tensor is formed as a linear combination of k independent symmetric rank-one random tensors, referred to as spikes, with weights referred to as signal-to-noise ratios (SNRs). The entries of the vectors that determine the spikes are i.i.d. sampled from general probability distributions supported on bounded subsets of R. This work focuses on the problem of detecting the presence of these spikes, and establishes the phase transition of this detection problem for any fixed k ≥ 1. In particular, it shows that for a set of relatively low SNRs it is impossible to distinguish between the spiked and non-spiked Gaussian tensors. Furthermore, in the interior of the complement of this set, where at least one of the k SNRs is relatively high, these two tensors are distinguishable by the likelihood ratio test. In addition, when the total number of low-rank components, k, grows in the order o(N (p−2)/4 ), the problem exhibits an analogous phase transition. This theory for spike detection implies that recovery of the spikes by the minimum mean square error exhibits the same phase transition. The main methods used in this work arise from the study of mean field spin glass models. In particular, the thresholds for phase transitions are identified as the critical inverse temperatures distinguishing the high and low-temperature regimes of the free energies in the pure p-spin model. This section states the main results of this paper and provides the necessary mathematical background. Additionally, it reviews prior results and describes the structure of the rest of the paper, particularly the structure of the proofs of Theorems 1 -4. Section 2.1 defines the necessary terminology, especially, the distinguishability of two random tensors. Section 2.2 describes our main