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The Free Tangent Law

Wiktor Ejsmont, Franz Lehner

Abstract

Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function and related functions in free probability. To be specific, we show that the function $$ \frac{\tan z}{1-x\tan z} $$ of Carlitz and Scoville describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers.

The Free Tangent Law

Abstract

Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function and related functions in free probability. To be specific, we show that the function of Carlitz and Scoville describes the limit distribution of sums of free commutators and anticommutators and thus the free cumulants are given by the Euler zigzag numbers.

Paper Structure

This paper contains 32 sections, 14 theorems, 121 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notation and Terminology
  4. Free Independence
  5. Free Convolution and the Cauchy-Stieltjes Transform
  6. Free infinite divisibility
  7. Wigner semicircle law
  8. Even elements
  9. Convergence in distribution
  10. Random matrices
  11. Convergence in eigenvalues
  12. Noncrossing Partitions
  13. Free Cumulants
  14. Combinatorics of tangent numbers
  15. An elementary lemma
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $r,n \in {\mathbb N}$ and $i_1 < i_2 < \dots < i_r = n$ be given and let be the induced interval partition. Consider now random variables ${\mathnormal X}_1,\dots,{\mathnormal X}_n\in\mathcal{A}$. Then the free cumulants of the products can be expanded as follows:

Figures (4)

  • Figure 1: The spectral radius of the generalized free tangent laws
  • Figure 2: Density and histogram ($M=1000$, $N=1000$) of the free tangent law
  • Figure 3: Density and histogram ($M=1000$, $N=1000$) of the free zigzag law
  • Figure 4: Densities of the generalized free tangent laws depending on $\alpha$

Theorems & Definitions (30)

  • Theorem 2.1
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • Remark 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 20 more