Concentration of measure, typical quantum states with fixed mean energy, and emergence of Gibbs states

The talk consists of two parts. In a first part, I give a brief elementary introduction to the "concentration of measure" phenomenon, a topic on the boundary between high-dimensional geometry and probability theory. This part also contains an overview on applications in quantum information theory and statistical mechanics. In the second part, I report on work in progress with David Gross and Jens Eisert. We investigate the properties of random pure quantum states under energy constraints, and show that with overwhelmingly large probability, they look in some sense like Gibbs states. Technically, we prove a measure concentration result on energy submanifolds.