Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2004-01-16
Physical Review Letters v. 92 (2004), 240601; Erratum: ibid. v.93 (2004),149901
Physics
Condensed Matter
Disordered Systems and Neural Networks
Updated version, close to the published variant, but with a misprint in Eq.(15) corrected
Scientific paper
10.1103/PhysRevLett.93.149901
Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in random environment. For N >>1 our asymptotic analysis reveals a phase transition at some critical value \mu_c of a control parameter \mu from a phase with finite landscape complexity to the phase with vanishing complexity. The same value of the control parameter is known to correspond to an onset of glassy behaviour at zero temperature. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.
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