Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices

Finding the mean of the total number Nₜₒₜ of stationary points for N-dimensional random energy landscapes is reduced to averaging the absolute value of the 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 a random environment. For N1 our asymptotic analysis reveals a phase transition at some critical value μc of a control parameter μ from a phase with a finite landscape complexity: Nₜₒₜ~e^NΣ, Σ(μ<μc)>0 to the phase with vanishing complexity: Σ(μ>μc)=0. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.