Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices

Finding the mean of the total number N(tot) 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 N>1 our asymptotic analysis reveals a phase transition at some critical value mu(c) of a control parameter mu from a phase with a finite landscape complexity: N(tot) approximately e(N Sigma), Sigma(mu0 to the phase with vanishing complexity: Sigma(mu>mu(c))=0. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.