1. Introduction

Noncommutative probability theory, a branch of mathematics born in the early 1980s, has emerged as a rich and captivating field that bridges the gap between conventional probability theory and its noncommutative counterpart. Before the advent of noncommutative probability theory, traditional probability theory provided the framework for understanding randomness and uncertainty in various fields. However, it had limitations when applied to settings where multiplication of random variables was noncommutative. This research gap prompted Dan Voiculescu to explore new avenues in probability theory. It has found applications in various fields, including operator algebras, random matrix theory, and combinatorics. Its significance lies in its ability to provide a unified framework for analyzing noncommutative random variables and solving problems that were previously intractable using classical probability theory.

In this review, we set the stage by providing a brief overview of the historical background, outlining the research topic, elucidating the research methods employed in this review, and emphasizing the significance of noncommutative probability theory as a field of study. The exploration is primarily based on a comprehensive literature review, encompassing seminal works, notable research papers, and specialized texts in the field of noncommutative probability theory. Our review focuses on the development of noncommutative probability theory, highlighting the key contributions of Dan Voiculescu, George Gheorghiu, Roland Speicher, Vic Reiner, and other notable mathematicians. We delve into the concepts of free independence, random matrices and non-crossing partitions within this framework.

2. Noncommutative probability theory

Noncommutative probability theory extends the concepts of classical probability theory to noncommutative algebras, primarily operator algebras. Classical probability theory deals with random variables on a commutative algebra. In contrast, noncommutative probability deals with the setting where random variables do not commute, a scenario also frequently encountered in quantum mechanics and other areas of mathematical physics.

The primary objects of study in classical probability are the distribution of random variables, which are functions from a sample space to the real line (or sometimes to more complex structures like the complex plane). Two random variables are said to be independent if their joint distribution is the product of their marginal distributions. Regarding to the noncommutative probability, instead of functions on a sample space, the primary objects of interest are self-adjoint elements of a noncommutative algebra, often an operator algebra. This algebra can represent observables in quantum mechanics, where the non-commutativity arises naturally.

The concept of noncommutative probability was introduced in the mid-1980s, when the Romanian-American mathematician Dan-Virgil Voiculescu made a significant contribution to the field of mathematics while researching the classification problem of von Neumann algebras. His work introduced a novel concept known as “free independence” within the realm of non-commutative probability spaces. This innovative idea marked the beginning of a profound connection between traditional probability theory and its non-commutative counterpart, where many core concepts and essential theorems found counterparts with both distinctions and connections.

In standard probability theory, concepts like independence find their counterparts in non-commutative probability spaces as free independence. Similarly, the familiar Gaussian distribution is mirrored by the semi-circular distribution, Gaussian random variables correspond to semi-circular elements, and the central limit theorem in conventional probability theory aligns with the central limit theorem in non-commutative probability spaces.

Initially, research in non-commutative probability spaces primarily centered on analytical methods, as evidenced by the researches done by Voiculescu, and had strong ties to operator theory [1-2]. However, as the 1990s unfolded, Voiculescu expanded upon his earlier work, developing the theory of free entropy in a series of publications, while Ge successfully applied these concepts to the study of operator algebras [3, 4]. Other specialized texts focusing on the analytical aspects of free probability theory are also achieved [5]. Over time, various branches of general probability theory, including stochastic analysis, have witnessed extensive development within non-commutative probability spaces.

The year 1991 marked a significant milestone when Voiculescu first bridged the gap between free probability theory and random matrices [6]. He introduced the concept of asymptotic freeness for random matrices and leveraged the methods of free probability theory to investigate the asymptotic freeness of specific random matrices. Subsequently, this area of research gained considerable momentum, with active contributions from researchers. And researches that emphasize the connection between free probability theory and random matrices are also established [7].

In 1994, Speicher established the combinatorial foundation of free probability theory [8]. He discovered that the definition of free independence in non-commutative probability spaces, the proof of the central limit theorem, and the computation of various distributions of random elements could all be systematically addressed through a class of specialized combinatorial structures known as non-crossing partitions. These non-crossing partitions allowed for the transformation of moments of random elements into cumulants, providing a clearer and more coherent framework for understanding free probability theory, such as the research of fundamental theory of cumulants in non-commutative probability spaces [9]. This area of research, focusing on combinatorial aspects of non-commutative probability spaces, continues to thrive.

The concept of “non-crossingness” emerged as a fundamental characteristic of virtually all combinatorial objects within non-commutative probability spaces. Non-crossing partitions, in particular, gained prominence as a well-behaved class of combinatorial structures with rich content, attracting decades of research attention. They established extensive connections with various algebraic and combinatorial objects, most notably symmetric groups or permutation groups. In the mid-1990s, P. Biane introduced a partial order relation on the symmetric group, effectively integrating non-crossing partitions into the Cayley graphs of symmetric groups. This integration facilitated the successful application of methods from free probability theory to investigate the asymptotic properties of Young diagrams in symmetric group representations [10]. Furthermore, non-crossing partitions within free probability theory sometimes adopt the language of non-crossing permutations, particularly in the study of high-order freeness of random matrices [11]. This demonstrates that the initially combinatorially defined concept of “non-crossingness,” rooted in order relations, can also be expressed in algebraic terms. The results presented in Chapters Three and Four of this paper provide additional support for this viewpoint.

In late 1990s, Reiner delved into the study of hyperplane arrangements and introduced three types of non-crossing partitions that corresponded to the three infinite families within Coxeter groups: A, B, and D types [12]. The classical combinatorial theory of non-commutative probability spaces predominantly relied on A-type non-crossing partitions. Biane and collaborators defined B-type non-commutative probability spaces based on B-type non-crossing partitions and explored their fundamental combinatorial properties in 2003 [13]. In 2009, Belinschi and Shlyakhtenko introduced the concept of infinitesimal freeness while investigating the analytical properties of B-type non-commutative probability spaces, paving the way for the development of infinitesimal non-commutative probability spaces by Fevrier and Nica [14-15]. The Infinitesimal non-commutative probability spaces stand out due to the unique feature that their random elements possess not only moments but also first-order formal derivatives of these moments. On one hand, they can be viewed as both a simplification and an extension of B-type non-commutative probability spaces. On the other hand, by unifying moment functions and their formal derivatives through coefficient transformations, the fundamental combinatorial techniques from A-type non-commutative probability spaces can be applied to infinitesimal non-commutative probability spaces.

The combinatorial structures of A, B, and D types introduced by Reine [16] have found application across various algebraic and combinatorial objects associated with Coxeter groups. Generally, the generalization from A-type to B-type is considered natural and exhibits well-behaved properties, while extension to D-type is relatively challenging. In the specific context of combinatorial structures related to non-commutative probability theory, and the work done by Goulden, Nica and Oancea have already successfully extended non-crossing partitions from the circular domain to B-type [17].

In 2007, Dykema introduced a unique class of combinatorial structures known as non-crossing linked partitions while exploring T-transformations within free probability theory [18]. These structures emerged as a fresh focus in the combinatorial research of non-commutative probability spaces and found applications in S-transformations and beyond. Therefore, investigating their combinatorial properties became imperative. Moreover, it is demonstrated that the combinatorial structures of B-type non-crossing linked partitions maintain close ties with algebraic structures [19-21].

3. Conclusion

In conclusion, this review article aimed to provide an overview of noncommutative probability theory, highlighting its historical development, key concepts, and significant combinatorial structures. However, like any field of study, noncommutative probability theory has its limitations and open questions. For instance, the extension from A-type to D-type non-crossing partitions remains a challenging problem, leave future endeavors could profitably focus on this potential generalization. Additionally, the nexus between non-commutative probability spaces and other mathematical realms, such as operator theory, could be further explored. Moreover, while infinitesimal non-commutative probability spaces offer promising insights, their full potential is yet to be explored.

To further advance this field, future research should focus on elucidating the properties and applications of D-type non-crossing partitions, exploring the intricate connections between various combinatorial structures, and uncovering new applications in mathematics and beyond. Noncommutative probability theory continues to be a fertile ground for exploration, promising exciting developments and deeper insights into the foundations of probability in non-commutative settings.