Complexity of the p-spin Hamiltonian with a Non-Rotationally Invariant Potential

TL;DR

The paper addresses the complexity of the pure $p$-spin Hamiltonian with a non-rotationally invariant polynomial potential by deriving a variational formula for the exponential growth rate of the mean number of critical points. It develops a rigorous Kac-Rice framework for an anisotropic Gaussian field, combined with truncation, Wegner estimates, and a Matrix Dyson Equation perspective to control determinant terms and spectral data. A joint large deviation principle for a tilted spectral-empirical pair is established, yielding a sharp variational representation $\Sigma(u)=\sup\{\mathcal{I}(\mu): \mu\in\mathcal{P}_{2q_2-2}(\mathbb{R})\cap\mathfrak{D}(u)\}$. From this representation, the paper derives finiteness and a finite critical energy level $u_c$ with $\limsup_{N\to\infty} u_N \le u_c$, hence providing an upper bound on the ground-state energy. The results extend the complexity program beyond isotropic, rotationally invariant settings and illuminate how polynomial confinement interacts with anisotropic Gaussian fields to shape the energy landscape.

Abstract

We investigate the complexity of the Hamiltonian in the pure $p$-spin spin glass model accompanied with a polynomial-type potential on $\mathbb{R}^N$. In this Hamiltonian, the Gaussian field is anisotropic, and the potential lacks rotational invariance. Our main result derives the logarithmic limit for the expected number of critical points in terms of a variational formula. As a consequence, by identifying the critical location of the phase transition from our representation, we provide an upper bound for the ground state energy of the model.

Theorems & Definitions (95)

• Remark 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 1.5
• Example 1.6
• Proposition 3.1
• Proposition 3.2
• proof
• Proposition 3.3: Conditional Distribution of $H_N$ and $\nabla^2H_N$ on $\nabla H_N$
• proof