Modulus of continuity of Kerov transition measure for continual Young diagrams

TL;DR

This paper investigates the stability of Kerov's transition measure, a key tool in asymptotic representation theory, by quantifying how small changes in continual Young diagrams affect their associated probability measures.

Contribution

It provides a quantitative analysis of the modulus of continuity for the homeomorphism between continual Young diagrams and their Kerov transition measures.

Findings

Established bounds on the modulus of continuity

Demonstrated stability of the transition measure under diagram perturbations

Extended understanding of the measure's dependence on diagram profiles

Abstract

The transition measure is a foundational concept introduced by Sergey Kerov to represent the shape of a Young diagram as a centered probability measure on the real line. Over a period of decades the transition measure turned out to be an invaluable tool for many problems of the asymptotic representation theory of the symmetric groups. Kerov also showed how to expand this notion for a wider class of continual diagrams so that the transition measure provides a homeomorphism between a subclass of continual diagrams (having a specific support) and a class of centered probability measures with a support contained in a specific interval. We quantify the modulus of continuity of this homeomorphism. More specifically, we study the dependence of the cumulative distribution function of Kerov transition measure on the profile of a diagram at the locations where the profile is not too steep.