Non-Commutative Probability and Related Topics 2025

Reference No.	2025a016
Type/Category	Grant for General Research-Workshop（Ⅱ）
Title of Research Project	Non-Commutative Probability and Related Topics 2025
Principal Investigator	Nobuhiro ASAI（Aichi University of Education・Professor）
Research Period	October 8,2025. - October 10,2025.
Keyword(s) of Research Fields	Noncommutative (quantum) probability, Random matrix, Mathematical physics, Learning theory
Abstract for Research Report	Noncommutative (quantum) probability theory is closely related to various mathematical disciplines, including mathematical physics, operator algebras, probability theory, combinatorics, function theory, quantum walks, and the theory of orthogonal polynomials. In recent years, this field has seen significant advancements. In Japan, in particular, practical applications have been actively explored, such as the analysis of deep learning using random matrices and the study of statistical properties of infinite-particle systems. This research aims to refine the methodologies of noncommutative probability theory and explore its potential applications in mathematical physics and information science. In this study, we seek to further develop the framework of noncommutative probability theory by conducting a detailed analysis of probability distributions and moment structures. Specifically, building on the four existing notions of independence (tensor independence, free independence, Boolean independence, and monotone independence), we characterize new types of probability distributions. Using techniques from combinatorics and function theory, we derive new limit theorems in noncommutative probability theory and clarify the asymptotic behavior of the corresponding limit distributions. Through these results, we aim to deepen our understanding of the moment structures and asymptotic properties of random variables, thereby advancing the theoretical foundation of noncommutative probability theory. Furthermore, by extending free probability theory in finite-dimensional matrix probability spaces and analyzing the behavior of random matrices, we explore the relationship between random matrices and probability distributions. This will help to establish a theoretical framework that unifies noncommutative probability theory and random matrix theory, broadening their applications. Additionally, to strengthen the connection with mathematical physics, we will conduct spectral analysis of random matrix models and quantum walk models using techniques from operator algebras and free probability theory. By doing so, we aim to deepen the linkage with mathematical physics and investigate the spectral properties of random matrices and the distribution characteristics of entanglement. This research seeks to establish new mathematical foundations that contribute significantly to applications in mathematical physics and information science. Through this research, we expect to expand the theoretical framework of noncommutative probability theory and further explore its connections with mathematical physics and information science. In particular, theoretical insights that contribute to improving the performance of machine learning in the field of AI represent a key challenge. More broadly, we aim to identify new problems based on the currently established theories and advance discussions toward their resolution. Additionally, fostering interactions among researchers, including early-career researchers, and promoting interdisciplinary discussions are also essential objectives of this study.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Nobuhiro ASAI（Aichi University of Education・Professor） Yuki UEDA（Hokkaido University of Education, Asahikawa・Associate Professor） Fumio HIROSHIMA（Kyusyu University・Professor） Hiroaki YOSHIDA（Josai University・Specially Appointed Professor） Etsuo SEGAWA（Yokohama National University・Professor）