# Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices

## TL;DR

This paper introduces random matrix-valued multiplicative functions, extending classical multiplicative functions to matrices, and analyzes their moments using linear recurrences and spectral radius bounds.

## Contribution

It generalizes Rademacher multiplicative functions to matrices and derives asymptotic second moments and bounds for higher moments based on spectral properties.

## Key findings

- Asymptotic second moment formula for matrix-valued functions
- Upper bounds for higher even moments
- Linear recurrence relations for Hilbert-Schmidt norms

## Abstract

We introduce the notion of a random matrix-valued multiplicative function, generalizing Rademacher random multiplicative functions to matrices. We provide an asymptotic for the second moment based on a linear recurrence property for Hilbert-Schmidt norms of sucessive products of random matrices. Moreover, we provide upper bounds for the higher even moments related to the generalized joint spectral radius.