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

TL;DR

This paper analyzes the complexity of a non-rotationally invariant p-spin Hamiltonian with polynomial potential, deriving a variational formula for critical points and providing bounds on the ground state energy.

Contribution

It introduces a variational approach to quantify the critical points and ground state energy in anisotropic p-spin models with non-invariant potentials.

Findings

Derived the logarithmic limit for the expected number of critical points.

Provided an upper bound for the ground state energy.

Identified the phase transition point in the model.

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.