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Hardcover, xviii+371 pages, NOT ex-library. Clean and bright throughout, unmarked text, no inscriptions/stamps, firmly bound. Minor shelfwear. -- An integration of quantum probability theory with spectral graph analysis, developing a powerful algebraic framework to investigate the asymptotic spectral behavior of large and growing graphs. This monograph builds upon von Neumann's foundational ideas, redefining random variables and measures in terms of self-adjoint operators and traces, and extends them into a structured analytic approach that reveals how quantum probabilistic tools can resolve longstanding questions in graph theory, operator algebras and mathematical physics. The core innovation is the method of quantum decomposition, which translates classical probability distributions into the language of interacting Fock spaces using orthogonal polynomials and three-term recurrence relations. By treating adjacency matrices as algebraic random variables, the authors develop a comprehensive method to derive spectral distributions and their limits in infinite graph sequences. This approach yields a unified framework in which quantum central limit theorems QCLTs take center stage, providing exact descriptions of asymptotic spectral distributions across families of distance-regular graphs. Chapters 1 & 2 establish foundational concepts in algebraic and quantum probability, including moment problems, Stieltjes transforms and Fock space constructions, before introducing the adjacency matrix as a noncommutative observable. The method is then applied across multiple graph classes. In homogeneous trees, the Wigner semicircle law and free Poisson distribution emerge naturally; in Hamming and Johnson graphs, Gaussian, exponential, geometric and Poisson limits are derived. Odd graphs lead to the two-sided Rayleigh distribution, demonstrating the framework's range. These results bridge classical and free probability in a concrete setting, offering a systematic toolkit for analyzing complex graph spectra. The book's discussion of quantum notions of independence (tensor, free, Boolean, monotone) is significant. By modeling these through graph operations such as the comb and star products, the authors provide new perspectives on the underlying algebraic structures and also practical methods for constructing graph models associated with different statistical behaviors. This graphical realization of independence deepens the connection between quantum probability and graph theory and sets the stage for novel applications in network science, random matrix theory and high-dimensional combinatorics. Chapters 9-12 pivot to the asymptotic representation theory of symmetric groups, using the tools developed to study Young diagrams under the Plancherel and Jack measures. The book offers a fresh derivation of the limit shape and Gaussian fluctuations in the Plancherel setting, connecting the spectral properties of adjacency matrices with Kerov's central limit theorem and its deformations. The authors also analyze the alpha-deformation via the Jack measure and demonstrate its implications for the Metropolis algorithm, revealing the practical relevance of their method in probabilistic sampling and statistical mechanics. By blending abstract algebraic constructions with detailed worked examples and asymptotic results, this volume addresses a broad range of contemporary mathematical problems. It contributes to ongoing discussions in noncommutative probability, spectral asymptotics, graph growth models and the analytic structure of symmetric group representations. Its accessible treatment makes it valuable for mathematicians and theoretical physicists working in quantum information, large-scale network analysis, random walks on groups or noncommutative harmonic analysis. It opens new avenues for applying quantum probabilistic methods to data science, particularly in contexts where complex network structures demand tools beyond classical probabilistic models.
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