# Joint spectrum and large deviation principle for random matrix products

## TL;DR

This paper explores the asymptotic behavior of random matrix products and introduces a limit set for powers of subsets in semisimple Lie groups, extending classical large deviation principles to matrix norms.

## Contribution

It establishes a large deviation principle for norms of random matrix products and defines a new limit set describing the asymptotic shape of powers in semisimple Lie groups.

## Key findings

- Large deviation principle for random matrix products.
- Introduction of a limit set for powers in semisimple Lie groups.
- Applications to the study of large deviations in linear groups.

## Abstract

The aim of this note is to announce some results about the probabilistic and deterministic asymptotic properties of linear groups. The first one is the analogue, for norms of random matrix products, of the classical theorem of Cramer on large deviation principles (LDP) for sums of iid real random variables. In the second result, we introduce a limit set describing the asymptotic shape of the powers of a subset S of a semisimple linear Lie group G (e.g. SL(d;R)). This limit set has applications, among others, in the study of large deviations.