Modulus of continuity of Kerov transition measure for continual Young diagrams

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.