Triviality of the geometry of mixed $p$-spin spherical Hamiltonians with external field

TL;DR

This paper proves that strong external fields simplify the landscape of mixed p-spin spherical Hamiltonians to only two critical points, contrasting with the complex landscape without or weak external fields.

Contribution

It establishes an explicit threshold for external field strength that guarantees trivial geometry in mixed p-spin models, extending previous results to more general cases.

Findings

Strong external fields lead to only two critical points.

Explicit threshold $h_c$ for trivialization is provided.

The results contrast with the complex landscape in weak or no external field cases.

Abstract

We study isotropic Gaussian random fields on the high-dimensional sphere with an added deterministic linear term, also known as mixed p-spin Hamiltonians with external field. We prove that if the external field is sufficiently strong, then the resulting function has trivial geometry, that is only two critical points. This contrasts with the situation of no or weak external field where these functions typically have an exponential number of critical points. We give an explicit threshold $h_c$ for the magnitude of the external fieldnecessary for trivialization and conjecture $h_c$ to be sharp. The Kac-Rice formula is our main tool. Our work extends [Fyo15], which identified the trivial regime for the special case of pure p-spin Hamiltonians with random external field.