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

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.

💡 Research Summary

The authors study a novel variant of the pure p‑spin glass Hamiltonian in which the underlying Gaussian field is anisotropic and the external potential V is a non‑rotationally invariant polynomial on ℝ^N. The Hamiltonian is \