Superconcentration and Related Topics
Series: Springer Monographs in Mathematics;
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Product details:
- Edition number 2014
- Publisher Springer
- Date of Publication 22 January 2014
- Number of Volumes 1 pieces, Book
- ISBN 9783319038858
- Binding Hardback
- No. of pages156 pages
- Size 235x155 mm
- Weight 454 g
- Language English
- Illustrations IX, 156 p. 0
Categories
Short description:
Based on the author's lectures at Cornell Probability Summer School in 2012, this book links the concept of superconcentration with probability theory. Includes a number of open problems for professional mathematicians and exercises for graduate students.
MoreLong description:
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.
MoreTable of Contents:
Preface.- 1.Introduction.- 2.Markov semigroups.- 3.Super concentration and chaos.- 4.Multiple valleys.- 5.Talagrand?s method for proving super concentration.- 6.The spectral method for proving super concentration.- 7.Independent flips.- 8.Extremal fields.- 9.Further applications of hypercontractivity.- 10.The interpolation method for proving chaos.- 11.Variance lower bounds.- 12.Dimensions of level sets.- Appendix A. Gaussian random variables.- Appendix B. Hypercontractivity.- Bibliography.- Indices.
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