Convergence for noncommutative rational functions evaluated in random matrices

One of the main applications of free probability is to show that for appropriately chosen independent copies of d random matrix models, any noncommutative polynomial in these d variables has a spectral distribution that converges asymptotically and can be described with the help of free probability. This paper aims to show that this can be extended to noncommutative rational functions, answering an open question by Roland Speicher. This paper also provides a noncommutative probability approach to approximating the free field. At the algebraic level, its construction relies on the approximation by generic matrices. On the other hand, it admits many embeddings in the algebra of operators affiliated with a I I-1 factor. A consequence of our result is that, as soon as the generators admit a random matrix model, the approximation of any self-adjoint noncommutative rational function by generic matrices can be upgraded at the level of convergence in distribution.

Automated summary

This paper investigates the convergence of the spectral distribution of noncommutative rational functions in independent random matrix models using free probability. It answers an open question by Roland Speicher and demonstrates that the approximation of self-adjoint noncommutative rational functions by generic matrices can be upgraded in terms of distribution convergence.