The Limit Shape Problem for Ensembles of Young Diagrams

? This book treats ensembles of Young diagrams originating from group-theoretical contexts and investigates what statistical properties are observed there in the large scale limit. We mainly focus on analyzing the interesting phenomenon that specific curves appear in 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, we develop asymptotic theory of representations 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. Our central destination 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. This book grows out of the notes of the lectures the author delivered at University of Wroclaw in Poland in 2007. Although the theme and flavor of the notes are maintained, we reconstruct and update the considerable contents in a unified manner.