Title Superconcentration and related topics / Sourav Chatterjee

Published New York : Springer, [2014] ©2014

Table of Contents
1.	Introduction					1
1.	Superconcentration					1
2.	Chaos					7
3.	Multiple Valleys					11
2.	Markov Semigroups					15
1.	Semigroup Basics					15
2.	The Ornstein-Uhlenbeck Semigroup					17
3.	Connection with Malliavin Calculus					18
4.	Poincare Inequalities					18
5.	Some Applications of the Gaussian Poincare Inequality					19
6.	Fourier Expansion					20
3.	Superconcentration and Chaos					23
1.	Definition of Superconcentration					23
2.	Superconcentration and Noise-Sensitivity					25
3.	Definition of Chaos					25
4.	Equivalence of Superconcentration and Chaos					27
5.	Some Applications of the Equivalence Theorem					28
6.	Chaotic Nature of the First Eigenvector					29
4.	Multiple Valleys					33
1.	Chaos Implies Multiple Valleys: The General Idea					33
2.	Multiple Valleys in Gaussian Polymers					34
3.	Multiple Valleys in the SK Model					36
4.	Multiple Peaks in Gaussian Fields					38
5.	Multiple Peaks in the NK Fitness Landscape					40
5.	Talagrand's Method for Proving Superconcentration					45
1.	Hypercontractivity					45
2.	Talagrand's L1-L2 Bound					47
3.	Talagrand's Method Always Works for Monotone Functions					49
4.	The Benjamini-Kalai-Schramm Trick					51
5.	Superconcentration in Gaussian Polymers					54
6.	Sharpness of the Logarithmic Improvement					56
6.	The Spectral Method for Proving Superconcentration					57
1.	Spectral Decomposition of the OU Semigroup					58
2.	An Improved Poincare Inequality					60
3.	Superconcentration in the SK Model					60
7.	Independent Flips					63
1.	The Independent Flips Semigroup					63
2.	Hypercontractivity for Independent Flips					65
3.	Chaos Under Independent Flips					66
8.	Extremal Fields					73
1.	Superconcentration in Extremal Fields					73
2.	A Sufficient Condition for Extremality					78
3.	Application to Spin Glasses					79
4.	Application to the Discrete Gaussian Free Field					83
9.	Further Applications of Hypercontractivity					87
1.	Superconcentration of the Largest Eigenvalue					87
2.	A Different Hypercontractive Tool					90
3.	Superconcentration in Low Correlation Fields					92
4.	Superconcentration in Subfields					93
5.	Discrete Gaussian Free Field on a Torus					94
6.	Gaussian Fields on Euclidean Spaces					99
10.	The Interpolation Method for Proving Chaos					105
1.	A General Theorem					105
2.	Application to the Sherrington-Kirkpatrick Model					111
3.	Sharpness of the Interpolation Method					112
11.	Variance Lower Bounds					115
1.	Some General Tools					115
2.	Application to the Edwards-Anderson Model					118
3.	Chaos in the Edwards-Anderson Model					119
12.	Dimensions of Level Sets					125
1.	Level Sets of Extremal Fields					125
2.	Induced Dimension					128
3.	Dimension of Near-Maximal Sets					131
4.	Applications					134
Appendix A	Gaussian Random Variables					137
1.	Tail Bounds					137
2.	Size of the Maximum					137
3.	Integration by Parts					139
4.	The Gaussian Concentration Inequality					139
5.	Concentration of the Maximum					140
Appendix B	Hypercontractivity					143
	References					147
	Author Index					153
	Subject Index					155

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Description 1 online resource (ix, 156 pages) : illustrations Series Springer monographs in mathematics, 2196-9922 Springer monographs in mathematics. 1439-7382 Contents Introduction -- Markov semigroups -- Superconcentration and chaos -- Multiple valleys -- Talagrand's method for proving superconcentration -- The spectral method for proving superconcentration -- Independent flips -- Extremal fields -- Further applications of hypercontractivity -- The interpolation method for proving chaos -- Variance lower bounds -- Dimensions of level sets Summary "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"--Publisher's description Bibliography Includes bibliographical references and index Notes Online resource; title from electronic title page (EBSCOHost, viewed March 2, 2018) Subject Probabilities. Mathematical statistics. Probability probability. MATHEMATICS -- Applied. MATHEMATICS -- Probability & Statistics -- General. Mathematical statistics Probabilities Form Electronic book LC no. 2014930163 ISBN 9783319038865 3319038869

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