# The central limit theorem for eigenvalues

## TL;DR

This paper proves a central limit theorem for the spectral radius of certain random walks on groups, showing that the distribution of eigenvalues' moduli converges to a normal distribution under specific conditions.

## Contribution

It establishes a CLT for eigenvalues of random walks on reductive groups, extending previous results to broader classes under minimal moment assumptions.

## Key findings

- Spectral radius follows a normal distribution asymptotically.
- Results apply to Zariski-dense random walks on reductive groups.
- Minimal moment conditions are sufficient for the CLT.

## Abstract

We prove that the spectral radius of a strongly irreducible random walk on GLd(R) (or more generally the vector of moduli of eigenvalues of a Zariski-dense random walk on a reductive group) satisfies a central limit theorem under an order two moment assumption.