On the anti-commutator of two free random variables

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-01-01 DOI:10.1512/iumj.2023.72.9505

Daniel Perales

引用次数: 2

Abstract

Let $(\kappa_n(a))_{n\geq 1}$ denote the sequence of free cumulants of a random variable $a$ in a non-commutative probability space $(\mathcal{A},\varphi)$. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants $(\kappa_n(ab+ba))_{n\geq 1}$ in terms of $(\kappa_n(a))_{n\geq 1}$ and $(\kappa_n(b))_{n\geq 1}$, where $a$ and $b$ are freely independent. Our formula expresses the $n$-th free cumulant of $ab+ba$ as a sum indexed by partitions in the set $\mathcal{Y}_{2n}$ of non-crossing partitions of the form

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两个自由随机变量的反对易子

设$(\kappa_n(a))_{n\geq 1}$表示一个随机变量$a$在非交换概率空间$(\mathcal{A},\varphi)$中的自由累积量序列。基于对二部图的一些考虑，我们提供了一个公式来计算$(\kappa_n(a))_{n\geq 1}$和$(\kappa_n(b))_{n\geq 1}$的累积量$(\kappa_n(ab+ba))_{n\geq 1}$，其中$a$和$b$是自由独立的。我们的公式将$ab+ba$的$n$ -自由累积量表示为按以下形式的非交叉分区集合$\mathcal{Y}_{2n}$中的分区索引的和

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.