# Linear eigenvalue statistics of random matrices with a variance profile

## TL;DR

This paper establishes bounds on the distribution of linear eigenvalue statistics of random matrices with variance profiles, proving a central limit theorem and re-establishing known fluctuation results using a second order Poincaré inequality.

## Contribution

It introduces a new upper bound on the total variation distance for eigenvalue statistics of variance-profile matrices and proves a general CLT for these statistics.

## Key findings

- Bound on total variation distance to Gaussian distribution
- Central limit theorem for matrices with various variance profiles
- Re-establishment of known fluctuation results

## Abstract

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centred, of a random matrix with a variance profile and the standard Gaussian random variable. The second order Poincar\'e inequality type result is used to establish the bound. Using this bound we prove Central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate variance profiles.