Publications Mathématiques de l'Institut des Hautes Études Scientifiques

, Volume 81, Issue 1, pp 73–205

Concentration of measure and isoperimetric inequalities in product spaces

  • Michel Talagrand

DOI: 10.1007/BF02699376

Cite this article as:
Talagrand, M. Publications Mathématiques de l’Institut des Hautes Scientifiques (1995) 81: 73. doi:10.1007/BF02699376


The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, “most” of the points of Ωn are “close” to A. We proceed to a systematic exploration of this phenomenon. The meaning of the word “most” is made rigorous by isoperimetrictype inequalities that bound the measure of the exceptional sets. The meaning of the work “close” is defined in three main ways, each of them giving rise to related, but different inequalities. The inequalities are all proved through a common scheme of proof. Remarkably, this simple approach not only yields qualitatively optimal results, but, in many cases, captures near optimal numerical constants. A large number of applications are given, in particular to Percolation, Geometric Probability, Probability in Banach Spaces, to demonstrate in concrete situations the extremely wide range of application of the abstract tools.

AMS Classification numbers

Primary 60E15, 28A35, 60G99Secondary 60G15, 68C15

Copyright information

© Publications Mathématiques de L’I.H.É.S. 1995

Authors and Affiliations

  • Michel Talagrand
    • 1
  1. 1.Equipe d’Analyse - Tour 48 UA au CNRS no 754Université Paris VIParis Cedex 05