[Paper Review] Eta-series and a Boolean Bercovici-Pata bijection for bounded k-tuples

Serban T. Belinschi, Alexandru Nica|ArXiv.org|Aug 25, 2006

Random Matrices and Applications9 references4 citations

TL;DR

This paper establishes a multivariate Boolean Bercovici–Pata bijection for bounded k-tuples of non-commutative random variables by introducing an η-series transform that linearizes free additive convolution. It proves that the R-transforms of infinitely divisible distributions coincide with the η-series of all bounded k-tuple distributions, enabling a bijection B: D_c(k) → D_c^{inf-div}(k) satisfying R_B(μ) = η_μ, and further shows that this bijection intertwines free multiplicative convolution via η_μ×ν = η_μ ⋆ η_ν.

ABSTRACT

On the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a $C^*$-probability space $D_c(k)$, one has an operation $\freeplus$ of free additive convolution, and one can consider the subspace $D_c^{inf-div}$ of distributions which are infinitely divisible with respect to this operation. The linearizing transform for free additive convolution is the R-transform. Thus, one has $R_{μ\freeplusν}=R_μ+R_ν$. The eta-series $η_μ$ is the counterpart of $R_μ$ in the theory of Boolean convolution. We prove that the space of eta-series of distributions belonging to $D_c(k)$ coincides with the space of R-transforms of distributions which are infinitely divisible with respect to free additive convolution. As a consequence of this fact, one can define a bijection $B : D_c(k) o D_c^{inf-div}$ via the formula $R_{B(μ)} = η_μ$, for all distributions $μ$ in $D_c(k)$. We show that $B$ is a multi-variable analogue of a bijection studied by Bercovici and Pata for k=1, and we prove a theorem about convergence in moments which parallels the Bercovici-Pata result. On the other hand we prove the formula $B(μ\freetimesν) = B(μ) \freetimes B(ν),$ with $μ,ν$ considered in a space $D^{alg}(k)$ containing $D_c (k)$ where the operation of free multiplicative convolution $\freetimes$ always makes sense. An equivalent reformulation for this equality is that $η_{μ\freetimesν}=η_μ \freestar η_ν,$ for all $μ,ν\in D^{alg}(k)$. This shows that, in a certain sense, eta-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

Motivation & Objective

• To extend the univariate Boolean Bercovici–Pata bijection to the multivariate setting for bounded k-tuples of non-commutative random variables.
• To characterize the set of R-transforms of infinitely divisible distributions as the image of η-series under a new transform.
• To establish a bijection B: D_c(k) → D_c^{inf-div}(k) such that R_B(μ) = η_μ, generalizing the univariate case.
• To prove that this bijection intertwines free multiplicative convolution, showing η_μ×ν = η_μ ⋆ η_ν in the algebraic framework D_alg(k).

Proposed method

• Define the η-series η_μ = M_μ / (1 + M_μ), where M_μ is the moment series of distribution μ ∈ D_c(k).
• Prove that the set of R-transforms of infinitely divisible k-tuple distributions equals the set of η-series of all bounded k-tuple distributions.
• Construct the bijection B: D_c(k) → D_c^{inf-div}(k) via the condition R_B(μ) = η_μ.
• Establish the compatibility of B with free multiplicative convolution by proving R_B(μ×ν) = R_B(μ) ⋆ R_B(ν), which is equivalent to η_μ×ν = η_μ ⋆ η_ν.
• Use combinatorial identities involving non-crossing partitions and Kreweras complements to verify the convolution identity for the ⋆-product of η-series.
• Leverage the linearizing property of the R-transform and the duality between R-transform and η-series to derive the main functional equations.

Experimental results

Research questions

• RQ1 Can the Boolean Bercovici–Pata bijection be generalized from the univariate to the multivariate setting for bounded k-tuples of non-commutative random variables?
• RQ2 Is the set of R-transforms of infinitely divisible distributions in D_c^{inf-div}(k) precisely the set of η-series of all distributions in D_c(k)?
• RQ3 Does the bijection B: D_c(k) → D_c^{inf-div}(k) defined by R_B(μ) = η_μ preserve the structure of free multiplicative convolution?
• RQ4 How do the η-series behave under free multiplicative convolution, and is there a natural operation on η-series that mirrors the R-transform's behavior under ×?
• RQ5 Can the combinatorial structure of non-crossing partitions be used to prove the identity η_μ×ν = η_μ ⋆ η_ν for μ, ν ∈ D_alg(k)?

Key findings

• The set of R-transforms of infinitely divisible k-tuple distributions equals the set of η-series of all bounded k-tuple distributions, i.e., {R_μ | μ ∈ D_c^{inf-div}(k)} = {η_μ | μ ∈ D_c(k)}.
• The map B: D_c(k) → D_c^{inf-div}(k) defined by R_B(μ) = η_μ is a well-defined bijection, generalizing the univariate Boolean Bercovici–Pata map.
• The bijection B intertwines free multiplicative convolution: B(μ × ν) = B(μ) × B(ν) holds for all μ, ν ∈ D_alg(k).
• The identity η_μ×ν = η_μ ⋆ η_ν holds for all μ, ν ∈ D_alg(k), where ⋆ is the operation on series defined by Nica and Speicher.
• The proof relies on a combinatorial identity involving non-crossing partitions: N′(σ, τ) = N′′(σ, τ), which is shown to be equivalent to the condition τ ≤ K(σ) and P_τ(n) = P_σ^{-1}(1).
• The transform Reta, defined by Reta(f) = η_μ when f = R_μ, satisfies Reta(f ⋆ g) = Reta(f) ⋆ Reta(g), establishing a homomorphism property for the ⋆-product.