  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 $II_1$ 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.
