Analytic aspects of the bi-free partial R-transform

TL;DR

This paper introduces a harmonic analysis approach to the bi-free R-transform in bi-free probability, focusing on commuting faces and extending classical limit theorems to the bi-free setting.

Contribution

It develops a harmonic analysis framework for the bi-free R-transform, enabling analysis of measures with unbounded support and establishing a bi-free analogue of classical limit laws.

Findings

Harmonic analysis approach to bi-free R-transform

Extension of limit theorems to bi-free infinitely divisible laws

Treatment of measures with unbounded support in bi-free context

Abstract

Since Voiculescu introduced his bi-free probability theory in 2013, the major development of the theory has been on its combinatorial side; in particular, on the combinatorics of bi-free cumulants and its application to the bi-free R-transform. In this article we propose a harmonic analysis approach to the bi-free R-transform, which is solely based on integral transforms of two variables. To accommodate the harmonic analysis tools, we confine ourselves in the simplest situation of bi-freeness with commuting faces. Our method allows us to treat measures with unbounded support, and we show that the classical limit theory of infinitely divisible laws, due to Levy and Khintchine, has a perfect bi-free analogue.