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

Understand

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.