Topics in concentration of measure

Classical concentration of measure gives powerful general techniques for bounding fluctuations of complicated random objects. However, these bounds do not always match the correct order of fluctuations. In more and more modern examples coming from a variety of areas including random matrices, polymer models, spin glasses, etc, this sub-optimality of the classical bounds is turning out to be more of a rule than an exception. We will discuss how this deficiency of the classical theory has deep implications about the structure of the problems and some directions towards building a new theory.

Курс прочитан в рамках St. Petersburg School in Probability and Statistical Physics.