Hessian spectrum at the global minimum of high-dimensional random landscapes

Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random N >> 1 dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature mu > 0. Simple landscapes with generically a single minimum are typical for mu > mu(c), and we show that the Hessian at the global minimum is always gapped, with the low spectral edge being strictly positive. When approaching from above the transitional point mu = mu(c) separating simple landscapes from 'glassy' ones, with exponentially abundant minima, the spectral gap vanishes as (mu-mu(c))(2). For mu < mu(c) the Hessian spectrum is qualitatively different for 'moderately complex' and 'genuinely complex' landscapes. The former are typical for short-range correlated random potentials and correspond to one-step replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching mu(c) from below with a larger critical exponent, as (mu(c)-mu)(4). At the same time in the 'most complex' landscapes with long-ranged power-law correlations the replica symmetry is completely broken. We show that in that case the Hessian remains gapless for all values of mu < mu(c), indicating the presence of 'marginally stable' spatial directions. Finally, the potentials with logarithmic correlations share both 1RSB nature and gapless spectrum. The spectral density of the Hessian always takes the semi-circular form, up to a shift and an amplitude that we explicitly calculate.