The concentration of measure phenomenon / Michel Ledoux.

Format:

Book

Author/Creator:

Ledoux, Michel, 1958-

Series:

Mathematical surveys and monographs ; no. 89.

Mathematical surveys and monographs, 0076-5376 ; v. 89

Language:

English

Subjects (All):

Concentration functions.

Measure theory.

Probabilities.

Physical Description:

x, 181 pages : illustrations ; 27 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, 2001.

Contents:

1. Concentration Functions and Inequalities 1

1.2 Concentration functions 3

1.3 Deviation inequalities 5

1.4 Observable diameter 14

1.5 Expansion coefficient 15

1.6 Laplace bounds and infimum-convolutions 16

2. Isoperimetric and Functional Examples 23

2.1 Isoperimetric examples 23

2.2 Brunn-Minkowski inequalities 32

2.3 Semigroup tools 38

3. Concentration and Geometry 47

3.1 Spectrum and concentration 47

3.2 Spectral and diameter bounds 53

3.3 Levy families 55

3.4 Topological applications 57

3.5 Euclidean sections of convex bodies 60

4. Concentration in Product Spaces 67

4.1 Martingale methods 67

4.2 Convex hull approximation 72

4.3 Control by several points 79

4.4 Convex infimum-convolution 82

4.5 The exponential distribution 83

5. Entropy and Concentration 91

5.1 Logarithmic Sobolev inequalities and concentration 91

5.2 Product measures 97

5.3 Modified logarithmic Sobolev inequalities 101

5.4 Discrete settings 108

5.5 Covariance identities 114

6. Transportation Cost Inequalities 117

6.1 Information inequalities and concentration 117

6.2 Quadratic transportation cost inequalities 122

6.3 Transportation for product and non-product measures 126

7. Sharp Bounds on Gaussian and Empirical Processes 133

7.1 Gaussian processes 133

7.2 Bounds on empirical processes 138

7.3 Sharper bounds via the entropic method 142

8. Selected Applications 151

8.1 Concentration of harmonic measures 151

8.2 Concentration for independent permutations 155

8.3 Subsequences, percolation, assignment 159

8.4 The spin glass free energy 163

8.5 Concentration of random matrices 167.

Notes:

Includes bibliographical references (pages 171-179) and index.

ISBN:

0821828649

OCLC:

47237544