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 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.

Demonstrates that the interplay of group representations and probability theory seems to be one of the international trends in mathematics Provides connections for readers with other branches and hints for new research as Young diagrams appear in various aspects of mathematics and physics Is written in a mathematically rigorous way using the prerequisite materials and is accessible to students and non-experts Includes supplementary material: sn.pub/extras

Inhalt

1. Introduction.- 2. Prerequisite materials.- 2.1 representations of the symmetric group.- 2.2 free probability.- 2.3 ensembles of Young diagrams.- 3. Analysis of the KerovOlshanski algebra.- 3.1 polynomial functions of Young diagrams.- 3.2 Kerov polynomials.- 4. Static model.- 4.1 Plancherel ensemble.- 4.2 Thoma and other ensembles.- 5. Dynamic model.- 5.1 hydrodynamic limit for the Plancherel ensemble.