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Title of Journal: J Stat Phys

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Abbravation: Journal of Statistical Physics

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Springer US

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DOI

10.1016/0022-3913(84)90090-8

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1572-9613

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Replica Symmetry Breaking Condition Exposed by Ran

Authors: Yan V Fyodorov Ian Williams
Publish Date: 2007/09/11
Volume: 129, Issue: 5-6, Pages: 1081-1116
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Abstract

We start with a rather detailed general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N≫1dimensional Gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity Then we employ random matrix methods to calculate the density of stationary points as well as minima of the associated energy surface This is used to show that for a generic smooth concave confining potentials the condition of the zerotemperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape and the mean number of minima are exponential in N For such systems the annealed complexity of minima vanishes cubically when approaching the critical confinement whereas the cumulative annealed complexity vanishes quadratically Different behaviour reported in our earlier short communication Fyodorov et al in JETP Lett 85261 2007 was due to nonanalyticity of the hardwall confinement potential Finally for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points In particular we show that in the vicinity of critical confinement the saddlepoints with a positive annealed complexity must be close to minima as they must have a vanishing fraction of negative eigenvalues in the Hessian

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