Counting Stationary Points of Random Landscapes as a Random Matrix Problem

TL;DR

This paper links the problem of counting stationary points in high-dimensional Gaussian landscapes to random matrix theory, providing analytical solutions and revealing a phase transition to a glassy state.

Contribution

It introduces a novel reduction of the stationary point counting problem to random matrix characteristic polynomial averages and offers an exact solution for a simplified landscape model.

Findings

Reduction of stationary point counting to characteristic polynomial averaging

Analytical solution for a toy model landscape

Discovery of a phase transition to a glass-like state

Abstract

Finding the mean of the total number of stationary points for N-dimensional random Gaussian landscapes can be reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. First such a reduction is illustrated for a class of models describing energy landscapes of elastic manifolds in random environment, and a general method of attacking the problem analytically is suggested. Then the exact solution to the problem [Phys. Rev. Lett. v.92 (2004) 240601] for a class of landscapes corresponding to the simplest, yet nontrivial "toy model" with N degrees of freedom is described. Asymptotic analysis reveals a phase transition to a glass-like state of the matter.