Diluted banded random matrices: Scaling behavior of eigenfunction and spectral properties

TL;DR

This paper investigates the scaling behavior of eigenfunctions and spectral properties in diluted banded random matrices, revealing a universal scaling law for localization length and eigenvalue spacing.

Contribution

It introduces a new scaling law for the normalized localization length and spectral properties in diluted banded random matrices, linking them through a single parameter.

Findings

Normalized localization length follows the law β=x*/(1+x*)

Scaling parameter x* depends on matrix sparsity and size as (b_eff^2/N)^δ

Eigenvalue spacing distribution also scales with x*

Abstract

We demonstrate that the normalised localization length $\beta$ of the eigenfunctions of diluted (sparse) banded random matrices follows the scaling law $\beta=x^*/(1+x^*)$. The scaling parameter of the model is defined as $x^*\propto(b_{eff}^2/N)^\delta$, where $b_{eff}$ is the average number of non-zero elements per matrix row, $N$ is the matrix size, and $\delta\sim 1$. Additionally, we show that $x^*$ also scales the spectral properties of the model (up to certain sparsity) characterized by the spacing distribution of eigenvalues.