Second Order Freeness and Fluctuations of Random Matrices, III. Higher order freeness and free cumulants

带融合的无穷小自由性的统一方法

A Unified approach to Infinitesimal Freeness with Amalgamation

本文在算子值框架下统一刻画 OVI 自由性，证明其等价于 $2\ imes2$ 上三角矩阵代数上的自由独立性，并借助 OVI 累积量给出判据及自由加、乘卷积公式。

通过自由累积量构造乘积 *-概率空间

Construction of product $*$-probability spaces via free cumulants

本文以混合自由累积量消失作为自由性定义，利用自由累积量而非矩来构造 *-概率空间的自由乘积。

Real and quaternionic second-order free cumulants and connections to matrix cumulants

We present definitions for real and quaternionic second-order free cumulants, functions whose collective vanshing when applied to elements from different subalgebras is equivalent to the second-order real (resp.\ quaternionic) freeness of those subalgebras. We construct a poset related to the annular noncrossing partitions, and calculate the Möbius function, which may be used to compute the coefficients in the expression for the second-order free cumulants. We show the connection between second-order free cumulants and the topological expansion interpretation of matrix cumulants. This provides…

The Matsumoto-Yor property in free probability via subordination and Boolean cumulants

We study the Matsumoto-Yor property in free probability. We prove three characterizations of free-GIG and free Poisson distributions by freeness properties together with some assumptions about conditional moments. Our main tools are subordination and Boolean cumulants. In particular, we establish a new connection between additive subordination function and Boolean cumulants.

Operator-valued distributions: I. Characterizations of freeness

Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give …

Free Probability for Pairs of Faces I

We consider a notion of bi-freeness for systems of non-commutative random variables with two faces, one of left variables and another of right variables. This includes bi-free convolution operations, bi-free cumulants and the bi-free central limit.

Free nested cumulants and an analogue of a formula of Brillinger

We prove a free analogue of Brillinger's formula (sometimes called "law of total cumulance") which expresses classical cumulants in terms of conditioned cumulants. As expected, the formula is obtained by replacing the lattice of set partitions by the lattice of noncrossing set partitions and using and an appropriate notion of noncommutative nested products. As an application we reprove a characterization of freeness due to Nica, Shlyakhtenko and Speicher by Möbius inversion techniques, without recourse to the Fock space model for free random variables.

Wigner矩阵的无穷小自由性

Infinitesimal Freeness of Wigner Matrices

本文在实无穷小自由概率框架下，通过环形非交叉划分与有向图的组合关系，计算独立复Wigner矩阵的无穷小累积量，证明其渐近实无穷小自由性，并特别指出矩阵与其转置的渐近无穷小自由性。

Cumulants in Noncommutative Probability Theory III. Creation- and annihilation operators on Fock spaces

Fock space constructions give rise to natural exchangeable families and are thus well suited for cumulant calculations. In this paper we develop some general formulas and compute cumulants for generalized Toeplitz operators, notably for q-Fock spaces, previously considered by M. Anshelevich and A. Nica, and Fock spaces for characters of the infinite symmetric group, which where constructed by Bozejko and Guta. An expression for cumulants in terms of the cycle-cover polynomials of certain directed graphs is obtained in this case.

张量自由概率理论：渐近张量自由性与中心极限定理

Tensor free probability theory: asymptotic tensor freeness and central limit theorem

本研究推广了Voiculescu的渐近自由独立性，针对局部幺正不变的张量积结构随机矩阵提出了张量自由性概念，通过混合张量自由累积量的消失来定义联合分布，并建立了张量自由版本的中心极限定理。

自由概率中的Kummer分布及其刻画

The Kummer distribution in free probability, and its characterizations

该研究聚焦自由概率领域，引入Kummer分布的类似物，借助自由乘法卷积从属关系与布尔累积量，基于自由性及条件矩假设完成自由Kummer与自由Poisson分布的刻画。

有限自由累积量：乘积卷积、亏格展开与无穷小分布

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

本文给出了多项式乘积卷积的有限自由累积量的组合公式及其拓扑展开，揭示了有限自由概率与自由概率的联系，并利用该展开研究了多项式的无穷小分布，最后为多项式导数根分布与自由分数卷积幂的联系提供了新证明。

与非平凡表示相关的紧群上独立均匀变量的强渐近自由性

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

本文研究紧群表示论衍生的新型矩阵酉模型，证明非平凡标记对应不可约表示下的酉商群满足强渐近自由性，同时得到正交群版本，并开发四种具独立理论价值的新工具。

高阶张量自由卷积、半圆律、自由泊松律与R变换

Tensorial free convolution, semicircular, free Poisson and R-transform in high order

本文为紧支撑测度建立张量自由卷积，定义高阶半圆律与自由泊松律，证明Wishart型张量收敛，并引入R变换及首批例子。

在c自由随机变量中的多项式及其在自由去噪中的应用

Polynomials in $c$-free random variables with applications to free denoising

研究c自由随机变量的多项式分布，建立布尔累积量递推关系，用于计算矩生成函数和分布，并应用于确定条件期望和正交投影。