Serban Belinschi, Hari Bercovici, Yinzheng Gu, Paul Skoufranis
In this paper we study some analytic properties of bi-free additive convolution, both scalar and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove a result about atoms of bi-free additive convolutions.
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Characterization problems in free probability are studied here. Using subordination of free additive and free multiplicative convolutions we generalize some known characterizations in free probability to random variables with unbounded support. Using this technique we also prove a new characterization of distributions of free random variables. A similar technique is used to study Laha-Lukacs regressions for monotonically independent random variables.
We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily numerically implementable. We present two concrete applications of our method: the product of two free operator-valued semicircular elements and the calculation of the distribution of $dcd+d^2cd^2$ for scalar-valued $c$ and $d$, which are free. Comparision between the solution obtained by our methods and simulations of random matrices shows excellent agreement.
Multi-variable subordination distributions for free additive convolution
设k为正整数，D_k表示C ~*-概率空间中k元组的联合分布空间.本文研究了μ，ν \in D_k的“μ \boxplus ν关于ν的从属分布”的概念，其中\boxplus是D_k上的自由加卷积运算.在这项研究中使用的主要工具是联合分布和相关的运营商模型的R-变换的组合特性，与运营商的作用在整个Fock空间多变量从属原来有很好的关系，向\boxplus -无限可分性的发展过程中D_k，最近发现的Belinschi和尼卡（arXiv：0711.3787）。最值得注意的是，人们可以更好地了解这个过程与自由布朗运动的联系。
We realize the Belinschi-Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov-Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated t…
Monotone max-convolution and subordination functions for free max-convolution
研究通过证明单调独立自 adjoint算子的谱最大值分布与经典最大卷积一致，引入自由最大卷积的子ordination函数并证明其存在性和结构性质。
Operators related to subordination for free multiplicative convolutions
Voiculescu和Biane已经证明了分析从属性质对自由加法和乘法卷积成立。本文给出了自由乘卷积从属性的一种操作方法。这项研究是基于“自由度与从属”，或“s-自由独立”，和“正交独立”，最近推出的自由添加剂卷积的背景下的概念。特别是，我们介绍和研究相关的乘法卷积和构造相关的运营商，所谓的“从属运营商”和“从属分支”。利用正交独立性，我们得到了从属分支的分解和相关的分解的s-自由和自由乘法卷积。操作方法导致几种新类型的图形产品，称为“循环产品”，与不同的概念的独立性（单调，布尔，正交，S-自由）。我们还证明了在图的圈积和图的自由积上的有根“交替双回游动”的计数给出了相应的乘法卷积的矩。
Boolean Cumulants and Subordination in Free Probability
本文研究自由卷积的从属关系，证明特定条件期望仍为预解式，并利用布尔累积量变换公式推广了自由加性与乘性卷积的从属公式，从而实现分布的显式计算。
Free multiplicative convolution with an arbitrary measure on the real line
本文研究实线上任意测度与非负实线上任意测度的自由乘法卷积，通过构造子ordination函数和S变换，证明了S变换的乘法性，并应用于稳定分布的卷积恒等式，扩展了Belinschi-Nica半群并建立了自由乘法卷积的正则性性质。
Operator models and analytic subordination for operator-valued free convolution powers
本文研究了算子值自由卷积幂的理论，提出了一种新的解析证明，并展示了算子模型与自由卷积的等价性。
We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Frechet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson's selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: How can we calculate …
In his article "On the free convolution with a semicircular distribution," Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy-Stieltjes transform of the free additive convolution of a probability measure on the real line with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane's methods and results to free convolution powers of operator-valued distributions and to free con…
Analytic Subordination Consequences of Free Markovianity
利用简单算子值解析函数的概念，从自由增量合并情形下广义预解式的解析从属结果出发，得到了算子代数的自由马尔可夫三元组的几个一般解析从属结果.这包括在酉自由乘法增量的情况下Biane从属结果的推广。
Analytic Subordination for Free Compression
我们将自由差商余代数方法推广到自由概率中的自由压缩的情况下的解析从属。
Analytic subordination results in free probability from non-coassociative derivation-comultiplications
我们扩展Voiculescu的方法，分析从属通过余代数的自由差商的非coassociative导子commultiplications出现在自由概率论。我们得到新的证明Voiculescu的分析从属结果自由马尔可夫三元组，乘法的酉是自由的合并。
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convol…
The asymptotic behavios of free convolution
本文给出了实直线上概率测度的无穷小三角形阵列的自由可加卷积的极限定理的简化证明。这个结果首先由Chistyakov和Götze使用解析从属证明。
The existence of Voiculescu's subordination functions in the context of non-tracial operator-valued C*-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the subordination functions thus obtained also satisfy an appropriately modified form of subordination for conditional expectations.
On freely indecomposable measures
我们证明了一个概率测度不是一个非平凡的自由加性卷积，如果它把没有质量的区间，其端点是原子。对自由乘卷积也证明了类似的结果。证明使用解析从属关系。
We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $\mathcal{T}$ of the $N$-regular tree, we define the $\mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $\mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the ope…
Free Perpetuities I: Existence, Subordination and Tail Asymptotics
本文研究自由概率中仿射不动点方程的解，涵盖次临界与临界情形，证明临界时级数几乎一致收敛但无高阶矩，并利用从属技术得到尾部幂律渐近，类比经典Kesten定理。
Decompositions of the free additive convolution
本文介绍并研究了一类与单调卷积有关的概率测度卷积，称为正交卷积。利用这个卷积，我们得到了自由概率中紧支持的概率测度的自由加性卷积的交替分解。这些分解与相关从属函数的交替分解直接相关。特别是，他们允许我们计算免费的加法卷积的紧支持的措施，而不使用自由累积量或R-变换。在简单的情况下，相应的柯西变换表示为连分数得到一个自然的方式。此外，这种方法在卷积和与非交换概率中的独立性（自由、单调和布尔）的主要概念相关的乘积之间建立了明确的联系。最后，我们的结果导致了有根图的自由积的自然分解。
We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.
We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution μone can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map ηfrom B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever ηis completely positive, while the free additive convolution power is defined whenever η- 1 is completely positive (where 1 stands fo…
We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for…
In this paper, we study the partial bi-free $S$-transform of a pair $(a,b)$ of random variables, and the $S$-transform of the $2\times 2$ matrix-valued random variable $\left(\begin{matrix}a&0\\0&b\end{matrix}\right)$ associated with $(a,b)$ when restricted to upper triangular $2\times 2$ matrices. We first derive an explicit expression of bi-free multiplicative convolution (of probability measures on the bi-unit-sphere $\mathbb{T}^2$ of $\mathbb{C}^2$, or on $\mathbb{R}^2_+$ in $\mathbb{C}^2$) from a subordination equation for bi-free multiplicative convolution. We then show that, when $(a_1,…
We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of i…
Limit theorems for additive c-free convolution
本文给出了测度对的无c卷积弱收敛的充要条件，其中测度是无穷小的，并且它们的支撑可以是无界的。这些结果是通过复分析方法得到的。
Limit laws for boolean convolutions
本文研究了无穷小三角形阵列中布尔独立随机变量的乘积及和的分布行为。我们证明了布尔卷积的极限律是由自由卷积的极限律决定的，反之亦然。我们进一步使用这些结果来显示经典卷积和布尔卷积的极限分布行为之间的几个连接。我们的结果的证明是基于分析仪器开发的自由卷积。
Free probability via entropic optimal transport
本文将加法、乘法、压缩及多变量自由运算的对数位势表述为熵最优传输变分问题，显式求解耦合并导出自由卷积与经典卷积之间的新不等式。
Free probability of type B: analytic interpretation and applications
本文给出了Biane、Goodman和Nica提出的B型自由卷积的解析解释，并给出了一个新的计算公式。这个公式使我们能够证明B型自由加法卷积本质上是条件自由卷积的重铸。我们证明了这种运算的几个方面，最重要的是它作为A型推导和自由卷积之间的“交织器”的出现。我们还显示了几个极限定理之间的联系，在类型A和类型B的自由概率。此外，我们表明，分析图片非常适合考虑B型随机变量的无穷小变形的想法，以普通的非交换随机变量。
On multiplicative conditionally free convolution
利用非交叉分区的组合学，我们构造了Voiculescu S变换的条件自由模拟。所得结果可用于条件自由乘性卷积的解析描述和无穷可除性的刻画。
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