Tensorial free convolution, semicircular, free Poisson and R-transform in high order

TL;DR

This paper extends free probability concepts to tensors, defining higher-order laws, establishing convergence results, and introducing a tensorial free convolution and R-transform for measures.

Contribution

It introduces tensorial analogues of semicircular and free Poisson laws, and develops a tensorial free convolution framework with an R-transform.

Findings

Convergence of Wishart-type tensors to free Poisson law

Convergence of Wigner tensors to semicircular law

Introduction of tensorial free convolution and R-transform

Abstract

This work builds on our previous developments regarding a notion of freeness for tensors. We aim to establish a tensorial free convolution for compactly supported measures. First, we define higher-order analogues of the semicircular (or Wigner) law and the free Poisson (or Marcenko-Pastur) law, giving their moments and free cumulants. We prove the convergence of a Wishart-type tensor to the free Poisson law and recall the convergence of a Wigner tensor to the semicircular law. We also present a free Central Limit Theorem in this context. Next, we introduce a tensorial free convolution, define the $R$-transform, and provide the first examples of free convolution of measures.