Simultaneous Asymptotics for the Shape of Random Young Tableaux with Growingly Reshuffled Alphabets

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 associated random Young tableaux are investigated, when both n and m converge together to infinity. If m does not grow too fast and if the draws are uniform, 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 towards the Tracy-Widom distribution.