Hessian spectrum at the global minimum and topology trivialization of locally isotropic Gaussian random fields

We study the energy landscape near the ground state of a model of a single particle in a random potential with trivial topology. More precisely, we find the large dimensional limit of the Hessian spectrum at the global minimum of the Hamiltonian $X_N(x)+\frac{\mu }{2}\mathrm{\parallel }x{\mathrm{\parallel }}^2,x\in {\mathbb{R}}^N,$ when $\mu$ is above the phase transition threshold so that the system has only one critical point. Here $X_N$ is a locally isotropic Gaussian random field. The same idea is also applied to study the more general model of elastic manifold. In the replica symmetric regime, our results confirm the predictions of Fyodorov and Le Doussal made in 2018 and 2020 using the replica method.