A certain curious feature of random objects, introduced by the
author as "super concentration," and two related topics, "chaos"
and "multiple valleys," are highlighted in this book. Although
super concentration has established itself as a recognized feature
in a number of areas of probability theory in the last twenty years
(under a variety of names), the author was the first to discover
and explore its connections with chaos and multiple valleys. He
achieves a substantial degree of simplification and clarity in the
presentation of these findings by using the spectral approach.
Understanding the fluctuations of random objects is one of the
major goals of probability theory and a whole subfield of
probability and analysis, called concentration of measure, is
devoted to understanding these fluctuations. This subfield offers a
range of tools for computing upper bounds on the orders of
fluctuations of very complicated random variables. Usually,
concentration of measure is useful when more direct
problem-specific approaches fail; as a result, it has massively
gained acceptance over the last forty years. And yet, there is a
large class of problems in which classical concentration of measure
produces suboptimal bounds on the order of fluctuations. Here lies
the substantial contribution of this book, which developed from a
set of six lectures the author first held at the Cornell
Probability Summer School in July 2012.
The book is interspersed with a sizable number of open problems
for professional mathematicians as well as exercises for graduate
students working in the fields of probability theory and
mathematical physics. The material is accessible to anyone who has
attended a graduate course in probability.
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