Central Limit Theorem for tensor products of free variables

TL;DR

This paper proves a central limit theorem for tensor products of free variables, revealing that the limiting distribution is either a semi-circle law or an interpolation between semi-circle and classical convolutions, depending on the variables' moments.

Contribution

It extends the central limit theorem to tensor products of free variables, characterizing the limiting distributions based on their mean and variance.

Findings

Limiting law is semi-circle for centered variables.

Non-centered variables lead to a free interpolation between semi-circle and classical convolution.

Provides a new understanding of tensor products in free probability.

Abstract

We establish a central limit theorem for tensor product random variables $c_k:=a_k \otimes a_k$, where $(a_k)_{k \in \mathbb{N}}$ is a free family of variables. We show that if the variables $a_k$ are centered, the limiting law is the semi-circle. Otherwise, the limiting law depends on the mean and variance of the variables $a_k$ and corresponds to a free interpolation between the semi-circle law and the classical convolution of two semi-circle laws.