The concentration of measure phenomenon.

*(English)*Zbl 0995.60002
Mathematical Surveys and Monographs. 89. Providence, RI: American Mathematical Society (AMS). x, 181 p. (2001).

Let \((X,d)\) be a metric space equipped with a (Borel) probability measure \(\mu\). The concentration function \(\alpha_{(X,d,\mu)}=\alpha_\mu\) is then defined by
\[
\alpha_\mu(r) := \sup\{1-\mu(A_r) : \mu(A)\geq 1/2\} ,\quad r>0 ,
\]
where \(A_r:=\{x\in X : d(x,A)<r\}\) denotes the open \(r\)-neighborhood of the Borel set \(A\). Now the concentration of measure phenomenon says that in many important examples \(\alpha_\mu(r)\) tends to zero very rapidly. In other words, whenever a Borel set \(A\subseteq X\) satisfies \(\mu(A)\geq 1/2\), then even a small enlargement of \(A\) contains already almost all mass of \(\mu\). The classical example of such a phenomenon (due to P. Levy and E. Schmidt) is the unit sphere \(\mathbf{S}^n\subset\mathbb R^{n+1}\) with geodesic distance \(d\) and (normed) Lebesgue measure \(\sigma^n\). Here we have
\[
\alpha_{\sigma^n}(r)\leq e^{-(n-1)r^2/2} ,\quad r>0 ,
\]
which becomes extremely powerful for large \(n\). Thus the (surprising) phenomenon appears that for large \(n\) the measure \(\sigma^n\) is almost concentrated in a small neighborhood of the equator of \(\mathbf{S}^n\).

Another possibility to formulate the concentration of measure phenomenon is in the language of Lipschitz functions \(F:X\to\mathbb R\). If \(m_F\) denotes the median of \(F\) and \(\|F\|_{\text{ Lip}}\leq 1\), it follows that \[ \mu(\{|F-m_F|\geq r\})\leq 2 \alpha_\mu(r) , \] hence, if the phenomenon appears, then \(F\) attains for most \(x\in X\) only values near to \(m_F\).

The concentration of measure phenomenon was mainly put forward in the early seventies by V. Milman in the asymptotic theory of Banach spaces. V. D. Milman and G. Schechtman [Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics. 1200. Berlin etc.: Springer-Verlag (1986; Zbl 0606.46013)] summarized the state of research till ’85. Since that time the investigation about the phenomenon developed very rapidly. Several new important examples have been found and they were applied very successfully in such various areas as geometry, functional analysis, infinite-dimensional integration, discrete mathematics, complexity theory and probability theory.

The aim of the book under review is to present the main results and applications obtained during the past 15 years e.g. by M. Talagrand about concentration of product measures, by S. Bobkov, B. Maurey, V. Milman, M. Gromov, M. Ledoux and others. The book contains also problems which are tightly related to the concentration of measure phenomenon as the logarithmic Sobolev inequality or transportation cost problems. The most important applications as e.g. for Gaussian stochastic processes, for empirical processes, for harmonic measures, random permutations or for the spin glass free energy function are subject of the last two chapters.

Summing up, it was undoubtedly a necessary task to collect all the results about concentration of measure during past years in a monograph. The author did this very successfully and the book is an important contribution to the topic. It will surely influence the further research in this area considerably. The book is very well written, and it was a great pleasure for the reviewer to read.

Another possibility to formulate the concentration of measure phenomenon is in the language of Lipschitz functions \(F:X\to\mathbb R\). If \(m_F\) denotes the median of \(F\) and \(\|F\|_{\text{ Lip}}\leq 1\), it follows that \[ \mu(\{|F-m_F|\geq r\})\leq 2 \alpha_\mu(r) , \] hence, if the phenomenon appears, then \(F\) attains for most \(x\in X\) only values near to \(m_F\).

The concentration of measure phenomenon was mainly put forward in the early seventies by V. Milman in the asymptotic theory of Banach spaces. V. D. Milman and G. Schechtman [Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics. 1200. Berlin etc.: Springer-Verlag (1986; Zbl 0606.46013)] summarized the state of research till ’85. Since that time the investigation about the phenomenon developed very rapidly. Several new important examples have been found and they were applied very successfully in such various areas as geometry, functional analysis, infinite-dimensional integration, discrete mathematics, complexity theory and probability theory.

The aim of the book under review is to present the main results and applications obtained during the past 15 years e.g. by M. Talagrand about concentration of product measures, by S. Bobkov, B. Maurey, V. Milman, M. Gromov, M. Ledoux and others. The book contains also problems which are tightly related to the concentration of measure phenomenon as the logarithmic Sobolev inequality or transportation cost problems. The most important applications as e.g. for Gaussian stochastic processes, for empirical processes, for harmonic measures, random permutations or for the spin glass free energy function are subject of the last two chapters.

Summing up, it was undoubtedly a necessary task to collect all the results about concentration of measure during past years in a monograph. The author did this very successfully and the book is an important contribution to the topic. It will surely influence the further research in this area considerably. The book is very well written, and it was a great pleasure for the reviewer to read.

Reviewer: Werner Linde (Jena)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

28C15 | Set functions and measures on topological spaces (regularity of measures, etc.) |

28A35 | Measures and integrals in product spaces |

46B09 | Probabilistic methods in Banach space theory |

60E15 | Inequalities; stochastic orderings |

82B44 | Disordered systems (random Ising models, random SchrĂ¶dinger operators, etc.) in equilibrium statistical mechanics |