Asymptotics for random Young diagrams when the word length and alphabet size simultaneously grow to infinity

TL;DR

This paper studies the asymptotic behavior of random Young diagrams derived from words with increasing length and alphabet size, revealing connections to GUE spectra and Tracy--Widom distribution under certain conditions.

Contribution

It provides new asymptotic results for the shape of Young tableaux when both word length and alphabet size grow simultaneously, extending classical results to more general settings.

Findings

Limiting shape matches GUE spectrum under uniform distribution and controlled growth of alphabet size.

First row of Young tableau converges to Tracy--Widom distribution in non-uniform case.

Results connect combinatorial structures with random matrix theory in asymptotic regimes.

Abstract

Given a random word of size $n$ whose letters are drawn independently from an ordered alphabet of size $m$, the fluctuations of the shape of the random RSK Young tableaux are investigated, when $n$ and $m$ converge together to infinity. If $m$ does not grow too fast and if the draws are uniform, then the limiting shape is the same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities will ensure the convergence of the first row of the tableau toward the Tracy--Widom distribution.