Transportation of measure, Young diagrams and random matrices (Q1769781)

The paper contains seven sections which present a new approach to transportation inequalities and the concentration-of-measure phenomenon for certain Gibbs measures on phase spaces of high dimension. In Section 2 basic results about tranportation of measure are given. In Section 4 the convexity properties of total energy are considered. Also logarithmic Sobolev inequalities and concentration inequalities for the continuous Gibbs measure are deduced. These results are deduced from an abstract logarithmic Sobolev inequality for general convex cost functions that is presented in Section 3. In Section 5 the empirical distribution of row lengths of Young diagrams is considered. In Section 6 it is shown how a special probability measure does indeed arise for the joint eigenvalue distributions of Dyson's generalized orthogonal ensemble with potential function \(\log\Gamma(1+x)\). Using the concentration inequalities of Section 6, in Section 7 convergence properties for various correlation functions associated with the random matrices are obtained.