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

Jean-Christophe Breton, Christian Houdr\'e
Arxiv ID: 0812.3672Last updated: 6/8/2021
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.

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