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Boolean Cumulants and Subordination in Free Probability

Franz Lehner, Kamil Szpojankowski

Abstract

We study subordination of free convolutions. We prove that for free random variables $X,Y$ and a Borel function $f$ the conditional expectation $E_\varphi\left[ (z-X-f(X)Yf^*(X))^{-1}| X\right]$, is a resolvent again. This result allows explicit calculation of the distribution of $X+f(X)Yf^*(X)$. The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.

Boolean Cumulants and Subordination in Free Probability

Abstract

We study subordination of free convolutions. We prove that for free random variables and a Borel function the conditional expectation , is a resolvent again. This result allows explicit calculation of the distribution of . The main tool is a formula for conditional expectations in terms of Boolean cumulant transforms, generalizing subordination formulas for free additive and multiplicative convolutions.

Paper Structure

This paper contains 36 sections, 20 theorems, 171 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free independence
  4. Set partitions
  5. Another partial order on noncrossing partitions
  6. Free cumulants
  7. Boolean cumulants
  8. Relation between free and Boolean cumulants
  9. Generating functions
  10. Conditional expectations in von Neumann algebras
  11. Boolean cumulants and subordination functions
  12. Yet another formula for expectations of free random variables
  13. A formula for conditional expectations of resolvents
  14. Subordination for multiplicative and additive free convolutions
  15. Subordination for $X+f(X)Yf^*(X)$
  16. ...and 21 more sections

Key Result

Proposition \oldthetheorem

Let $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$ be mutually free families in a noncommutative probability space $(\mathcal{A},\varphi)$, then where in the above sum for each fixed sequence $0< i_1<\ldots< i_k<n$ we fix $i_0=0$ and $i_{k+1}=n$. Consequently, if $\mathcal{B}$ is a subalgebra containing $\{X_1,X_2,\dots,X_n\}$ and free from $\{Y_1,Y_2,\dots,Y_{n-1}\}$ then the conditional expectation

Figures (5)

  • Figure 1: Partition diagrams
  • Figure 2: A partition illustrating the proof of Lemma \ref{['prop:31']}: $\pi'$ is green and has four irreducible components, $\pi"$ is blue and red, the latter marking the distinguished componet $\pi_0$
  • Figure 3: Newton polygons of the three factors
  • Figure 4: The spectral density of $X+XYX$ for Wigner law
  • Figure 5: The spectral density of $X+XYX$ for arcsine law

Theorems & Definitions (38)

  • Proposition \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: $X$ and $Y$ exchanged
  • proof : Proof 1
  • proof : Sketch of a proof based on FMNS2
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 28 more