Free Probability for Pairs of Faces I

TL;DR

This paper introduces a framework called bi-freeness for analyzing systems of non-commutative random variables with two distinct faces, expanding free probability theory to include bi-free convolution, cumulants, and the central limit.

Contribution

It develops the concept of bi-freeness for pairs of faces, extending free probability to a new setting with left and right variables and their associated convolution and cumulant structures.

Findings

Defines bi-freeness for systems with two faces.

Establishes bi-free convolution operations.

Derives bi-free cumulants and the bi-free central limit theorem.

Abstract

We consider a notion of bi-freeness for systems of non-commutative random variables with two faces, one of left variables and another of right variables. This includes bi-free convolution operations, bi-free cumulants and the bi-free central limit.