"RANDOM MATRICES AND RELATED TOPICS" |
Monthly Colloquia |
Abstract: The distribution of the Young diagrams partitioning N, under the Plancherel measure, converges almost surely as N tends to infinity to the Vershik distribution. The proof presented in this paper uses transportation of measure and concentration phenomena to establish weak L^2 convergence and convergence of the associated distributions in the Wasserstein metric. Related matrix ensembles also satisfy Gaussian concentration phenomena for the joint eigenvalue distribution and the empirical eigenvalue distribution, with constants that improve as the matrix size increases. These results combine the mean field theory approach to generalized orthogonal ensembles pursued by Boutet De Monvel, Pastur and Shecherbina with the theory of displacement convexity in mass transport.