Eigenfunction Statistics of Complex Systems: A Common Mathematical Formulation

TL;DR

This paper presents a unified mathematical framework for eigenfunction statistics in complex systems, highlighting how system size and complexity influence eigenfunction behavior, unlike eigenvalue statistics which depend solely on complexity.

Contribution

It introduces a common multi-parametric probability density formulation for eigenfunctions, linking system properties to universality classes based on complexity.

Findings

Eigenfunction statistics depend on both system size and complexity parameter.

Eigenvalue statistics are sensitive only to the complexity parameter.

The formulation suggests possible classification of physical properties into universality classes.

Abstract

We derive a common mathematical formulation for the eigenfunction statistics of Hermitian operators, represented by a multi-parametric probability density. The system-information in the formulation enters through two parameters only, namely, system size and the complexity parameter, a function of all system parameter including size. The behavior is contrary to the eigenvalue statistics which is sensitive to complexity parameter only and shows a single parametric scaling. The existence of a mathematical formulation, of both eigenfunctions and eigenvalues, common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties. This also suggests the possibility to classify them in various universality classes defined by complexity parameter.