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

TL;DR

This paper investigates the complexity of high-dimensional random energy landscapes, revealing a phase transition related to glassy behavior, by analyzing the spectral properties of associated random matrices.

Contribution

It provides an exact solution for the mean number of critical points in a class of random landscapes and identifies a phase transition at a critical parameter value.

Findings

Exact solution for finite N landscapes

Identification of a phase transition at critical 

Connection between landscape complexity and glass transition

Abstract

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.