Hessian spectrum at the global minimum of the spherical pure-like mixed p-spin glasses

Abstract

We study the large $N$-dimensional limit of the Hessian spectrum at the global minimum of some subclasses of the spherical mixed $p$-spin models. Specifically, we show that its empirical spectral measure converges in probability to a shifted and rescaled semicircle law and does not have outliers. Our method follows the second moment approach developed recently in \cite{BSZ20}, from which the ground state energy can be derived for the $pure$-$like$ mixed $p$-spin model. By analyzing the complexity function with given radial derivative and energy, we derive the convergence of the Hessian spectrum from the vanishing mean number of critical points. For the $1$-RSB model, the ground state energy was explicitly computed in \cite{huang2023constructive}. Combined with the complexity function of local maxima with given radial derivative obtained in \cite{belius2022complexity}, this allows us to obtain the corresponding results more directly. Our result extends those corresponding results in the regime of topology trivialization.