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

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.