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

Yan V Fyodorov, Pierre Le Doussal

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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.
Original languageEnglish
Number of pages31
JournalJournal of Physics A
Early online date10 Oct 2018
Publication statusPublished - 26 Oct 2018


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