Concentration of measure and related topics

Au sujet de ce cours

The concentration of measure phenomenon lies at the heart of the analysis of high-dimensional stochastic models. Its most basic instance is the second moment method, or Chebyshev inequality. More sophisticated examples include concentration results for empirical spectral distributions and Talagrand concentration inequality for product measures and convex and Lipschitz functions.

Concentration of measure can be viewed as a quantitative or non-asymptotic counterpart to the central limit phenomenon and to large deviation principles. Also it is central to statistical physics more broadly. In this course, we
will mostly focus on a collection of concentration inequalities, and we will also explore selected links with the following topics:

convex geometry and isoperimetry
optimal transport
probabilistic functional analysis
random matrices and Coulomb/Riesz gases
randomized combinatorial optimization

This course will be taught at ENS. It is designed to be accessible without prerequisites on conditional expectation or stochastic processes. It is naturally connected to other M2 courses, without dependency or redundancy.

Bibliography

Anderson, Guionnet, Zeitouni – An introduction to random matrices (CUP 2010)
Boucheron, Lugosi, Massart – Concentration inequalities. A non asymptotic theory of independence (OUP 2016)
Ledoux – The concentration of measure phenomenon (AMS 2005)
Santambrogio – Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling (Birkhäuser 2015)
Van Handel – Probability in High Dimension (Princeton lecture notes, 2021)
Vershynin – High-Dimensional Probability, an introduction (CUP 2018)