Transportation of measure, Young diagrams and random matrices

The theory of transportation of measure for general convex cost functions is used to obtain a novel logarithmic Sobolev inequality, which leads to a transportation inequality and hence a concentration of measure inequality. There are applications to the Plancherel measure associ-ated to the symmetric group, the distribution of Young diagrams partitioning N as N!1 and to the mean eld theory of random matrices. For the potential log (x + 1), the gener-alized orthogonal ensemble and its empirical eigenvalue distribution are shown to satisfy a Gaussian concentration of measure phenomenon. Hence the empirical distribution converges weakly almost surely as the matrix size increases; the limiting density is the derivative of te Vershik probability density