Linear low energy excitations in fully-connected models of glasses

We study the linear excitations around typical energy minima of a mean-field disordered model with continuous degrees of freedom undergoing a random first order transition. Contrary to naive expectations, the spectra of linear excitations are ungapped and we find the presence of a pseudogap corresponding to localized excitations with arbitrary low excitation energy. Moving to deeper minima in the landscape, the excitations appear increasingly localized while their abundance decreases. Beside typical minima, there also exist rare ultra-stable minima, with an energy gap and no localised excitations.