Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Third Order Cumulants of products

Octavio Arizmendi, Daniel Munoz George, Saylé Sigarreta

TL;DR

The paper develops a third-order analogue of higher-order freeness, deriving a general formula for the third-order cumulants of products in terms of sums over the combinatorial class of non-crossing partitioned permutations $\mathcal{PS}_{NC}$ with a separability condition.The authors introduce and exploit the machinery of non-crossing partitioned permutations across three circles, extend several topological and ordering lemmas from the two-circle case, and prove the main theorem by induction on $(r,s,t)$.They then apply the main result to multiple random-matrix ensembles, including Gaussian Ginibre, Wishart models, and products of third-order free circular elements, obtaining explicit cumulant and moment expressions, and establish that third-order $R$-diagonality is preserved under multiplication by free factors.These results extend known second-order formulas to third order, providing concrete formulas for cumulants and fluctuation moments of products, with implications for the analysis of multi-factor random matrix products and related operator-algebraic structures.

Abstract

We provide a formula for the third order free cumulants of products as entries. We apply this formula to find the third order free cumulants of various Random Matrix Ensambles including product of Ginibre Matrices and Wishart matrices, both in the Gaussian case.

Third Order Cumulants of products

TL;DR

The paper develops a third-order analogue of higher-order freeness, deriving a general formula for the third-order cumulants of products in terms of sums over the combinatorial class of non-crossing partitioned permutations $\mathcal{PS}_{NC}$ with a separability condition.The authors introduce and exploit the machinery of non-crossing partitioned permutations across three circles, extend several topological and ordering lemmas from the two-circle case, and prove the main theorem by induction on $(r,s,t)$.They then apply the main result to multiple random-matrix ensembles, including Gaussian Ginibre, Wishart models, and products of third-order free circular elements, obtaining explicit cumulant and moment expressions, and establish that third-order $R$-diagonality is preserved under multiplication by free factors.These results extend known second-order formulas to third order, providing concrete formulas for cumulants and fluctuation moments of products, with implications for the analysis of multi-factor random matrix products and related operator-algebraic structures.

Abstract

We provide a formula for the third order free cumulants of products as entries. We apply this formula to find the third order free cumulants of various Random Matrix Ensambles including product of Ginibre Matrices and Wishart matrices, both in the Gaussian case.

Paper Structure

This paper contains 16 sections, 37 theorems, 214 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on noncrossing partitions, permutations and partitioned permutations
  3. The set of partitions and non-crossing permutations
  4. Non-crossing partitioned permutations.
  5. Notation
  6. Partial order and other relations on $S_n$
  7. Topology of non-crossing permutations
  8. Preliminary results
  9. Proof of main theorem
  10. Applications
  11. $s^2$
  12. $cac^*$
  13. Product with an R-diagonal
  14. Cumulants of $aa^*$ for an $R$-diagonal Operator.
  15. Product of circular elements
  16. ...and 1 more sections

Key Result

Theorem 1.1

where the summation is over those $(\mathcal{V},\pi)\in \mathcal{PS}_{NC}(p,q,l)$ such that $\pi^{-1}\gamma_{p,q,l}$ separates the points of $N \vcentcolon= \{n_1,n_1+n_2,\dots,n_1+\cdots +n_{r+s+t}\}$ and $p=n_1+\cdots+n_r$, $q=n_{r+1}+\cdots +n_{r+s}$ and $l=n_{r+s+1}+\cdots+n_{r+s+t}$.

Figures (6)

  • Figure 1: An example of a non-crossing partition in $\mathcal{NC}(6)$.
  • Figure 2: $(\mathcal{V},\pi)\in PS_{NC}^{(1)}(8,4,3)$.
  • Figure 3: Permutations in solid lines and partitions in dashed lines.
  • Figure 4: Representation of the permutations used in Example \ref{['e1']}.
  • Figure 5: Representation of the permutations used in Example \ref{['e2']}.
  • ...and 1 more figures

Theorems & Definitions (98)

  • Theorem 1.1: Third order cumulants with products as arguments
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Example 2.6
  • Remark 2.7
  • Lemma 2.8
  • Example 2.9
  • Remark 3.1
  • Example 3.3
  • ...and 88 more