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

Abstract

Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random $N\gg 1$ 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 {\it 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 1-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 {\it 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.

28 pages, 1 figure; a brief summary of main results is added to the introduction

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