The theory of transportation of measure for general convex cost functions is used to obtain a novel logarithmic Sobolev inequality for measures on phase spaces of high dimension and hence a concentration-of-measure inequality. There are applications to the Plancherel measure associated with the symmetric group, the distribution of Young diagrams partitioning N as N→∞ and to the mean-field theory of random matrices. For the potential logΓ(x+1), the generalized orthogonal ensemble and its empirical eigenvalue distribution are shown to satisfy a Gaussian concentration-of-measure phenomenon. Hence the empirical eigenvalue distribution converges weakly almost surely as the matrix size increases; the limiting density is given by the derivative of the Vershik probability density.