An analytic approach to the finite R-transform

TL;DR

This paper provides an analytic perspective on the finite R-transform, connecting it to Laplace transforms and proving convergence to free additive convolution with explicit error bounds.

Contribution

It introduces an analytic approach to the finite R-transform, establishing convergence results and error estimates for finite free additive convolution.

Findings

Finite R-transform differs from Voiculescu R-transform by O(N^{-1})

Established convergence of finite free additive convolution to free additive convolution

Connected finite free Fourier transform to Laplace transform through logarithmic potentials

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.