The purpose of this book is to introduce readers to certain topics in random matrix theory that specifically involve the phenomenon of concentration of measure in high dimension. Partly this work was motivated by researches in the EC network Phenomena in High Dimension, which applied results from functional analysis to problems in statistical physics. Pisier described this as the transfer of technology, and this book develops this philosophy by discussing applications to random matrix theory of:

1. (i) optimal transportation theory;

2. (ii) logarithmic Sobolev inequalities;

3. (iii) exponential concentration inequalities;

4. (iv) Hankel operators.

Recently some approaches to functional inequalities have emerged that make a unified treatment possible; in particular, optimal transportation links together seemingly disparate ideas about convergence to equilibrium. Furthermore, optimal transportation connects familiar results from the calculus of variations with the modern theory of diffusions and gradient flows.

I hope that postgraduate students will find this book useful and, with them in mind, have selected topics with potential for further development. Prerequisites for this book are linear algebra, calculus, complex analysis, Lebesgue integration, metric spaces and basic Hilbert space theory. The book does not use stochastic calculus or the theory of integrable systems, so as to widen the possible readership.

In their survey of random matrices and Banach spaces, Davidson and Szarek present results on Gaussian random matrices and then indicate that some of the results should extend to a wider context by the theory of concentration of measure [152].