# Finite-rank perturbations of random band matrices via infinitesimal free probability

## TL;DR

This paper investigates the infinitesimal spectral distribution of banded GUE matrices, establishing a phase transition at band width proportional to the square root of matrix size, and extends results on finite-rank perturbations and outlier detection.

## Contribution

It proves a sharp phase transition for the infinitesimal distribution of banded GUE matrices and extends infinitesimal free probability results to this setting.

## Key findings

- Sharp $\sqrt{N}$ transition for infinitesimal distribution
- Model is infinitesimally free from matrix units and all-ones matrix for large band widths
- Finite-rank perturbations produce outliers at classical positions

## Abstract

We prove a sharp $\sqrt{N}$ transition for the infinitesimal distribution of a periodically banded GUE matrix. For band widths $b_N = \Omega(\sqrt{N})$, we further prove that our model is infinitesimally free from the matrix units and the normalized all-ones matrix. Our results allow us to extend previous work of Shlyakhtenko on finite-rank perturbations of Wigner matrices in the infinitesimal framework. For finite-rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble.