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

2024/12/03 by Remi Bonnin, Bonnin, Remi · 3 citations
Engineering · #Advanced Numerical Analysis Techniques #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2412.02572

openalex publication_date 2024/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Characteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors
• The large N factorization does not hold for arbitrary multi-trace observables in random tensors
• Characterization of Gaussian Tensor Ensembles

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