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Infinitesimal freeness for orthogonally invariant random matrices

Guillaume Cébron, James A Mingo

Abstract

We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.

Infinitesimal freeness for orthogonally invariant random matrices

Abstract

We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real infinitesimal cumulants and show that real infinitesimal freeness is equivalent to vanishing of mixed cumulants. We prove the formula for cumulants with products as entries.

Paper Structure

This paper contains 19 sections, 24 theorems, 208 equations, 8 figures.

Table of Contents

  1. Introduction and Statement of Results
  2. Complex Infinitesimal Free Freeness
  3. Real Infinitesimal Probability Spaces
  4. Integration by parts for random matrices
  5. Asymptotic Freeness of Orthogonally Invariant Ensembles
  6. Real Infinitesimal Free Cumulants and the Moment-cumulant formula
  7. Symmetric annular non-crossing permutations
  8. Spatial Derivatives
  9. The Moment-Cumulant Formula
  10. Genus calculations
  11. Real infinitesimal cumulants and real infinitesimal freeness
  12. The product formula
  13. Small Examples with the Square of a Semi-Circle
  14. The proof of Theorem \ref{['thm:product rule']}: outline
  15. First Step: Proposition \ref{['prop:first']}
  16. ...and 4 more sections

Key Result

Proposition 3.4

Let $(\mathcal{A}, \tau, \tau', t)$ be a real infinitesimal probability space with $\tau$ and $\tau'$ tracial. Let $\mathcal{A}_1, \dots, \mathcal{A}_s \subseteq \mathcal{A}$ be unital symmetric subalgebras which are free with respect to $\tau$. Then $\mathcal{A}_1, \dots, \mathcal{A}_n$ are real in

Figures (8)

  • Figure 1: A non crossing permutation of a $(6, 4)$-annulus.
  • Figure 2: A symmetric non-crossing annular permutation on a $(6, -6)$-annulus. Note that the orientation of the points on the two circles is the same. This is the opposite convention used in Figure \ref{['fig:non-crossing_annular']}.
  • Figure 3: When $n = 3$ there are three elements in $S_\mathit{NC}^{\delta, a}(3, -3)$, they are displayed above.
  • Figure 4: When $n = 3$ there are six elements in $S_\mathit{NC}^{\delta}(3, -3)$; the first three are displayed in Figure \ref{['fig:all through\n blocks']}, the remaining three are displayed above.
  • Figure 5: The 5 non-crossing pairings of a $(4, -4)$-annulus mentioned in § \ref{['section:small example']}. We have marked the positions of the points $\{2, -2, 4, -4\}$ in the Kreweras complement. Only the third and the fifth have the property that $K^\delta(\pi)$ separates the points of $\{2, -2, 4, -4\}$. These are the two that contribute to $\kappa_2'(x, x)$. $\diamond$
  • ...and 3 more figures

Theorems & Definitions (53)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Remark 5.1
  • Lemma 5.2
  • ...and 43 more