# Non-commutative rational function in strongly convergent random variables

## TL;DR

This paper demonstrates that non-commutative rational functions can be extended to strongly convergent random variables, showing stability under inversion, with applications to GUE matrices converging to free semi-circular elements.

## Contribution

It establishes the extension of non-commutative rational functions to strongly convergent variables and proves stability of strong convergence under taking inverses.

## Key findings

- Strong convergence is stable under inversion.
- Rational functions of GUE matrices converge to free semi-circular elements.
- Almost sure convergence in trace and norm for rational functions of GUE matrices.

## Abstract

Random matrices like GUE, GOE and GSE have been studied for decades and have been shown that they possess a lot of nice properties. In 2005, a new property of independent GUE random matrices is discovered by Haagerup and Thorbj{\o}rnsen in their paper [18], it is called strong convergence property and then more random matrices with this property are followed (see [27], [5], [1], [24], [10] and [3]). In general, the definition can be stated for a sequence of tuples over some \text{C}^{\ast}-algebras. And in this general setting, some stability property under reduced free product can be achieved (see Skoufranis [30] and Pisier [26]), as an analogy of the result by Camille Male [24] for random matrices. In this paper, we want to show that, for a sequence of strongly convergent random variables, non-commutative polynomials can be extended to non-commutative rational functions under certain assumptions. Roughly speaking, the strong convergence property is stable under taking the inverse. As a direct corollary, we can conclude that for a tuple (X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)}) of independent GUE random matrices, r(X_{1}^{\left(n\right)},\cdots,X_{m}^{\left(n\right)}) converges in trace and in norm to r(s_{1},\cdots,s_{m}) almost surely, where r is a rational function and (s_{1},\cdots,s_{m}) is a tuple of freely independent semi-circular elements which lies in the domain of r.