Energy distribution of maxima and minima in a one-dimensional random system

TL;DR

This paper investigates how the energy distribution of maxima and minima in a one-dimensional disordered system depends on the correlation length of the disorder, revealing a transition from separated to mixed stationary points.

Contribution

It provides a detailed analysis of the energy distribution of stationary points in 1D disordered systems, highlighting the impact of disorder correlation length on maxima and minima.

Findings

Short-range correlated disorder leads to energy separation between maxima and minima.

Long-range correlated disorder results in maxima and minima being completely mixed.

The study characterizes the statistical properties of stationary points in disordered Hamiltonians.

Abstract

We study the energy distribution of maxima and minima of a simple one-dimensional disordered Hamiltonian. We find that in systems with short range correlated disorder there is energy separation between maxima and minima, such that at fixed energy only one kind of stationary points is dominant in number over the other. On the other hand, in the case of systems with long range correlated disorder maxima and minima are completely mixed.