1995/12/07 by Michel Talagrand · 10 citations
Mathematics · #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · doi:10.1007/bf02699376
crossref issued 1995/12/07 · crossref published 1995/12/07 · crossref published-online 1995/12/07 · openalex publication_date 1995/12/07 · crossref created 2007/09/02 · openalex created_date 2025/10/10 · crossref deposited 2026/02/17 · openalex updated_date 2026/08/01 · crossref indexed 2026/08/03
The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product Ω N of probability spaces has measure at least one half, “most” of the points of Ω n 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.