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 $X_N(x) +\frac\mu2 \|x\|^2, x\in\mathbb{R}^N,$ 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. In the regime
of topology trivialization, our results confirm in a strong sense the
prediction of Fyodorov and Le Doussal made in 2018 using the replica method.

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