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This book treats ensembles of Young diagrams originating from group-theoretical contexts and investigates what statistical properties are observed there in a large-scale limit. The focus is mainly on analyzing the interesting phenomenon that specific curves appear in the appropriate scaling limit for the profiles of Young diagrams. This problem is regarded as an important origin of recent vital studies on harmonic analysis of huge symmetry structures. As mathematics, an asymptotic theory of representations is developed of the symmetric groups of degree n as n goes to infinity. The framework of rigorous limit theorems (especially the law of large numbers) in probability theory is employed as well as combinatorial analysis of group characters of symmetric groups and applications of Voiculescu's free probability. The central destination here is a clear description of the asymptotic behavior of rescaled profiles of Young diagrams in the Plancherel ensemble from both static and dynamic points of view.

Mathematics. --- Group theory. --- Topological groups. --- Lie groups. --- System theory. --- Probabilities. --- Mathematical physics. --- Mathematical Physics. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Probability Theory and Stochastic Processes. --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Limit cycles. --- Limit cycles --- Cycles, Limit --- Differential equations --- Limit cycles of differential equations --- Physical mathematics --- Physics --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Systems, Theory of --- Systems science --- Science --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Math --- Philosophy --- Differentiable dynamical systems --- Topological Groups. --- Distribution (Probability theory. --- Statistical physics. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistical methods --- Dynamical systems. --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics