Focusing on ensembles of Young diagrams from group theory, this book explores their statistical properties in large-scale limits. It highlights the emergence of specific curves in the scaling limits of Young diagram profiles, linking this to significant studies in harmonic analysis of symmetry structures. The author develops an asymptotic theory for symmetric group representations as n approaches infinity, utilizing rigorous limit theorems and combinatorial analysis. The goal is to provide a comprehensive understanding of the asymptotic behavior of rescaled profiles in the Plancherel ensemble.
The new Springer Quantum Probability Programme is inaugurated by a monograph from Akihito Hora and Nobuaki Obata, showcasing key features of contemporary quantum probability. This work employs specific quantum probabilistic techniques to address longstanding problems with extensive existing literature, independent of quantum probability. It also establishes connections between seemingly disparate areas of mathematics, such as orthogonal polynomials, graph theory, Nevanlinna's theory, and the representation theory of the symmetric group. The focus on the asymptotic behavior of large graphs is increasingly relevant across various fields, including solid-state physics, complex networks, biology, telecommunications, operations research, and combinatorial optimization. This broad applicability extends the book's audience beyond mathematicians to include physicists, engineers, biologists, and economists. The sophisticated analytical tools used to analyze discrete structures like graphs are particularly intriguing. While the application of analysis to uncover properties of discrete structures has a rich tradition in number theory, its use in graph theory is relatively new and poised for significant growth. The book is unified by two main ideas of quantum probability, notably the quantum decomposition of classical random variables.
