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arXiv · 2605.02093

An analytic approach to the finite R-transform

Octavio Arizmendi·Katsunori Fujie
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Abstract

We revisit Marcus' finite free analogue of Voiculescu $R$-transform from an analytic viewpoint. By relating the finite free Fourier transform to the Laplace transform, we study the finite $R$-transform through logarithmic potentials and Legendre transforms. Under suitable assumptions, we prove that the finite $R$-transform of a polynomial differs from the Voiculescu $R$-transform of its empirical root distribution by $O(N^{-1})$. As an application, we obtain an analytic proof of the convergence of finite free additive convolution to free additive convolution.

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Octavio Arizmendi, Katsunori Fujie. 2026-05-03. An analytic approach to the finite R-transform. https://arxiv.org/abs/2605.02093

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