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Finding the mean of the total number N_{tot} of critical points forN-dimensional random energy landscapes is reduced to averaging the absolutevalue of characteristic polynomial of the corresponding Hessian. For any finiteN we provide the exact solution to the problem for a class of landscapescorresponding to the "toy model" of manifolds in random environment. For N >>1our asymptotic analysis reveals a phase transition at some critical value \mu_cof a control parameter \mu from a phase with finite landscape complexity to thephase with vanishing complexity. The same value of the control parameter isknown to correspond to an onset of glassy behaviour at zero temperature.Finally, we discuss a method of dealing with the modulus of the spectraldeterminant applicable to a broad class of problems.Comment: Updated version, close to the published variant, but with a misprint  in Eq.(15) correcte

physical review letters

Erratum: Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of Spectral Determinant of Random Matrices [Phys. Rev. Lett.<b>92</b>, 240601 (2004)]

Erratum: Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of Spectral Determinant of Random Matrices [Phys. Rev. Lett.92, 240601 (2004)]