Superconcentration and Related Topics

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Springer Science & Business Media, Jan 9, 2014 - Mathematics - 156 pages

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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Contents

Introduction
1
Markov Semigroups
15
Superconcentration and Chaos
23
Multiple Valleys
33
Talagrands Method for Proving Superconcentration
44
The Spectral Method for Proving Superconcentration
57
Independent Flips
63
Extremal Fields
73
The Interpolation Method for Proving Chaos
105
Variance Lower Bounds
115
Dimensions of Level Sets
124
Gaussian Random Variables
137
Hypercontractivity
143
References
146
Author Index
153
Subject Index
155

Further Applications of Hypercontractivity
86

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About the author (2014)

Sourav Chatterjee is a Professor of Statistics and Mathematics at Stanford University. He has previously taught at the University of California at Berkeley and at the Courant Institute of Mathematical Sciences. He has won several international awards for his work in probability theory, including the Rollo Davidson Prize (2010), the Doeblin Prize (2012) and the Loève Prize (2013) and has received the invitation to speak at the International Congress of Mathematicians in 2014.

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