On the energy landscape of the mixed even $p$-spin model

TL;DR

This paper analyzes the energy landscape of the mixed even p-spin model, revealing the existence of exponentially many configurations near any energy level, and establishes disorder chaos and fluctuation properties of the maximal energy.

Contribution

It improves previous results by showing exponential growth in the number of orthogonal peaks at maximal energy and proves disorder chaos at zero temperature for the model.

Findings

Existence of exponentially many configurations near any energy level.

Exponential number of orthogonal peaks at maximal energy.

Disorder chaos established at zero temperature and any external field.

Abstract

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.