The finite R-transform of a polynomial differs from the Voiculescu R-transform of its empirical root distribution by O(N^{-1}), providing an analytic proof that finite free additive convolution converges to free additive convolution.
Fujie, Regularity and convergence properties of finite free convolutions,Int
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An analytic approach to the finite R-transform
The finite R-transform of a polynomial differs from the Voiculescu R-transform of its empirical root distribution by O(N^{-1}), providing an analytic proof that finite free additive convolution converges to free additive convolution.