The density of stationary points in a high-dimensional random energy landscape and the onset of glassy behaviour

Abstract

We calculate the density of stationary points and minima of a $N\gg 1$ dimensional Gaussian energy landscape. We use it to show that the point of zero-temperature replica symmetry breaking in the equilibrium statistical mechanics of a particle placed in such a landscape in a spherical box of size $L=R\sqrt{N}$ corresponds to the onset of exponential in $N$ growth of the cumulative number of stationary points, but not necessarily the minima. For finite temperatures we construct a simple variational upper bound on the true free energy of the $R=\infty$ version of the problem and show that this approximation is able to recover the position of the whole de-Almeida-Thouless line.

a revised and shortened version with a few typos corrected and references added. To appear in JETP Letters

References in corpus (4)

• Large Deviations of Extreme Eigenvalues of Random Matrices
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• Marginal States in Mean Field Glasses
• Free energy landscapes, dynamics and the edge of chaos in mean-field models of spin glasses

Cited by in corpus (6)

• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Arrangement of nearby minima and saddles in the mixed spherical energy landscapes
• Counting metastable states of Ising spin glasses on arbitrary graphs
• When is the average number of saddle points typical?
• Density of critical points for a Gaussian random function