Limit shape for regularisation of large partitions under the Plancherel measure

TL;DR

This paper extends the classical asymptotic shape results for large partitions under the Plancherel measure to their e-regularisations, revealing a convex limit shape and providing explicit asymptotics for the largest rows and columns.

Contribution

It proves that e-regularisations of partitions under the Plancherel measure have a convex limit shape, a shaking of the classical curve, with explicit asymptotic formulas.

Findings

Convex limit shape for e-regularised partitions

Explicit asymptotics for first row and column lengths

Extension of classical shape results to regularised partitions

Abstract

A celebrated result of Kerov-Vershik and Logan-Shepp gives an asymptotic shape for large partitions under the Plancherel measure. We prove that when we consider $e$-regularisations of such partitions we still have a convex limit shape, which is given by a shaking of the Kerov-Vershik-Logan-Shepp curve. We deduce an explicit form for the first asymptotics of the length of the first rows and the first columns for the $e$-regularisation.