Statistical analysis of chiral structured ensembles: role of matrix constraints

TL;DR

This paper investigates how specific constraints on matrix elements influence the spectral and eigenfunction statistics of complex systems, revealing that eigenfunction behavior depends on relative strengths rather than the number of independent elements.

Contribution

It introduces a numerical analysis of constrained matrix ensembles, challenging the traditional one-to-one correlation between eigenvalue and eigenfunction statistics.

Findings

Spectral statistics are highly sensitive to the number of independent matrix elements.

Eigenfunction statistics depend primarily on the relative strengths of matrix elements.

Contradicts the conventional belief linking Poisson and Wigner-Dyson statistics to localization and delocalization.

Abstract

We numerically analyze the statistical properties of complex system with conditions subjecting the matrix elements to a set of specific constraints besides symmetry, resulting in various structures in their matrix representation. Our results reveal an important trend: while the spectral statistics is strongly sensitive to the number of independent matrix elements, the eigenfunction statistics seems to be affected only by their relative strengths. This is contrary to previously held belief of one to one relation between the statistics of the eigenfunctions and eigenvalues (e.g. associating Poisson statistics to the localized eigenfunctions and Wigner-Dyson statistics to delocalized ones).