Concentration of Measure (E16) Non-Examinable (Graduate Level)



Concentration of Measure

Nathanael Berestycki and Richard Nickl

The concentration of measure phenomenon was rst put forward in geometric functional analysis by Milman and Gromov, and has been subject to fascinating recent developments, particularly in probability theory. Roughly speaking, this phenomenon says that random variables in high or in nite-dimensional spaces tend to be \nearly constant". It can be quanti ed explicitly by so-called concentration inequalities.

The aim of this course is to investigate the basic mathematical principles behind the concentration of measure phenomenon. The arguments often rely on a combination of ideas from probability, geometry, analysis and statistics. This remarkable synthesis makes the subject both very elegant and powerful. We will then illustrate how to apply these results to some concrete examples. Topics to be covered include:

  • Poincare and isoperimetric inequalities; basic spectral geometry
  • Entropy and Log-Sobolev inequalities
  • Concentration of Gaussian measures (Borell's inequality)
  • Talagrand's inequality and sharp concentration inequalities for product measures, including applications to empirical processes
  • Sharp thresholds and the Kahn-Kalai-Linial theorem, including applications to rst-passage percolation

 

Desirable Previous Knowledge

We shall only assume some basic notions of probability and measure theory. This being a non-examinable course, we plan to make this as informal and accessible as possible.

Lecture notes

A draft set of lecture notes is available at http://www.statslab.cam.ac.uk/~beresty/teach/cm.html

A standard reference for part of this material is:
M. Ledoux (2001). The concentration of measure phenomenon. AMS monographs, Providence.

 

 

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