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

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.