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The Limit Shape Problem for Ensembles of Young Diagrams

  • Book
  • © 2016

Overview

Authors:
  1. Akihito Hora

    1. Department of Mathematics, Hokkaido University, Sapporo, Japan

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  • 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

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 17)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-ix
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  2. Preliminaries

    • Akihito Hora
    Pages 1-13

  3. Analysis of the Kerov–Olshanski Algebra

    • Akihito Hora
    Pages 15-30

  4. Continuous Diagram

    • Akihito Hora
    Pages 31-41

  5. Static Model

    • Akihito Hora
    Pages 43-60

  6. Dynamic Model

    • Akihito Hora
    Pages 61-68

  7. Back Matter

    Pages 69-73
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Hokkaido University, Sapporo, Japan

    Akihito Hora

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