Asymptotics of the major index of a random standard tableau

TL;DR

This paper investigates the asymptotic behavior of the major index in random standard tableaux, establishing convergence properties and probabilistic estimates as the tableau shape varies in the Thoma simplex.

Contribution

It introduces the mod-$$ convergence framework for the major index of random tableaux with shapes in the Thoma simplex, providing new probabilistic and large deviation results.

Findings

Proves mod-$$ convergence of the major index

Derives Berry--Esseen type convergence speed estimates

Establishes strong large deviation principles

Abstract

In this article, we establish the mod-$\phi$ convergence of the major index of a uniform random standard tableau whose shape converges in the Thoma simplex. This implies various probabilistic estimates, in particular speed of convergence estimates of Berry--Esseen type, and strong large deviation principles.