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.