Abstract
The density of stationary points and minima of a N ≫ 1 dimensional Gaussian energy landscape has been calculated. It is used 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 √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, a simple variational upper bound on the true free energy of the R = ∞ version of the problem has been constructed and it has been shown that this approximation can recover the position of the whole de-Almeida-Thouless line.
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Fyodorov, Y.V., Sommers, H.J. & Williams, I. Density of stationary points in a high dimensional random energy landscape and the onset of glassy behavior. Jetp Lett. 85, 261–266 (2007). https://doi.org/10.1134/S0021364007050098
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DOI: https://doi.org/10.1134/S0021364007050098