On Weighted Random Band-Matrices with Dependences

TL;DR

This paper develops methods to analyze the eigenvalue distribution of weighted random band matrices with dependent entries, establishing moment calculations and weak convergence results for these complex matrices.

Contribution

It introduces techniques to compute eigenvalue moments for matrices with dependent, weighted entries, extending analysis to band matrices with bandwidth o(N).

Findings

Derived formulas for eigenvalue moments with dependent entries.

Proved weak convergence of eigenvalue statistics for weighted and band matrices.

Extended spectral analysis to matrices with non-independent entries.

Abstract

We develop techniques to compute the k-th Moment of the Eigenvalue-statistic for a random Matrix M the entries of which do not have to be necessarily Independent. The dependence is controlled via an equivalence relation on the pairs of the entries of M. Furthermore the entries are weighted, that means they are multiplied with a Riemann-integrable function on (0,1). Finally also weak convergence in probability of the Eigenvalue statsitic is discussed not for the weighted Matrix but also for band matrices the band-width of which behaves like o(N).