Triviality of the Geometry of Mixed p-Spin Spherical Hamiltonians with External Field

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