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

Remi Bonnin

TL;DR

This work extends Voiculescu's freeness to tensors by defining a tensorial free convolution for compactly supported measures. It introduces higher-order semicircular and free Poisson laws with explicit moments and free cumulants, proves convergence results for Wigner and Wishart tensors, and establishes a tensorial additive convolution with an $R$-transform (and a $Q$-transform) that is additive under convolution. Basic examples illustrate how tensorial free convolution operates on the high-order laws and highlight connections to Fuss-Catalan and Fuss-Narayana combinatorics. The framework provides a foundation for high-dimensional tensor statistics and links combinatorial map invariants to free probabilistic convolution in the tensor setting.

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.

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

TL;DR

This work extends Voiculescu's freeness to tensors by defining a tensorial free convolution for compactly supported measures. It introduces higher-order semicircular and free Poisson laws with explicit moments and free cumulants, proves convergence results for Wigner and Wishart tensors, and establishes a tensorial additive convolution with an -transform (and a -transform) that is additive under convolution. Basic examples illustrate how tensorial free convolution operates on the high-order laws and highlight connections to Fuss-Catalan and Fuss-Narayana combinatorics. The framework provides a foundation for high-dimensional tensor statistics and links combinatorial map invariants to free probabilistic convolution in the tensor setting.

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 -transform, and provide the first examples of free convolution of measures.

Paper Structure

This paper contains 35 sections, 25 theorems, 106 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Maps of tensors.
  3. Non-crossing poset.
  4. Main results
  5. High order laws.
  6. Convergence results.
  7. Free convolution of tensors.
  8. Organization of the paper
  9. Acknowledgements
  10. Moments and free cumulants for tensors
  11. Maps of tensors
  12. Combinatorial maps.
  13. Tensor maps.
  14. Non-crossing poset
  15. Moments of a tensor
  16. ...and 20 more sections

Key Result

Theorem 1

We have as functional relation in $\mathbb {C}{{\left[ \space \left[ z \right] \space \right] }}$,

Figures (5)

  • Figure 1: Some tensor maps.
  • Figure 2: Poset $\mathcal{P}_{\pi}$
  • Figure 3: $T$ and $T^{\sigma}$.
  • Figure 4: A $\Delta$-graph ($p=4, n=3, b=2, r+1=5$).
  • Figure 5: A $\Delta_1(n,b)$-graph.

Theorems & Definitions (53)

  • Theorem 1: Analytic moment-cumulant formula
  • Theorem 2: Semicircular law in high order
  • Theorem 3: Free Poisson law in high order
  • Theorem 4: High order Wigner Theorem
  • Theorem 5: High order Marčenko-Pastur Theorem
  • Lemma 1
  • Theorem 6: Free CLT
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • ...and 43 more