The class of [concentration of measure inequalities][1] is a fundamental tool in modern probability (and any field that uses probability, e.g., random matrix theory, theoretical computer science, statistics, high-dimensional geometry, combinatorics, etc.). As explained in [this blog post of Scott Aaronson][2], these are basic ways in which one "upper bounds the probability of something bad", and often the bounds are exponential or even gaussian in nature when one is far away from the mean (or median) and there are many independent (or somewhat independent) variables involved. Examples of such inequalities include - The [Chernoff inequality][3] and its relatives ([Hoeffding][4], [Bernstein][5], [Bennett][6], etc.) - [Azuma's inequality][7] - [McDiarmid's inequality][8] - [Levy's inequality][9] - [Talagrand's concentration inequality][10] Log-Sobolev inequalites are, strictly speaking, not concentration of measure inequalities, but are often closely related to them, thanks to techniques such as the Herbst argument. A standard reference in the subject for these topics is <cite authors="Ledoux, Michel">_Ledoux, Michel_, The concentration of measure phenomenon, Mathematical Surveys and Monographs. 89. Providence, RI: American Mathematical Society (AMS). x, 181 p. (2001). [ZBL0995.60002](https://zbmath.org/?q=an:0995.60002).</cite> I also have a blog post on this topic [here][11]. [1]: https://en.wikipedia.org/wiki/Concentration_of_measure [2]: https://www.scottaaronson.com/blog/?p=3712 [3]: https://en.wikipedia.org/wiki/Chernoff_bound [4]: https://en.wikipedia.org/wiki/Hoeffding%27s_inequality [5]: https://en.wikipedia.org/wiki/Bernstein_inequalities_(probability_theory) [6]: https://en.wikipedia.org/wiki/Bennett%27s_inequality [7]: https://en.wikipedia.org/wiki/Azuma%27s_inequality [8]: https://en.wikipedia.org/wiki/McDiarmid%27s_inequality [9]: https://en.wikipedia.org/wiki/Concentration_of_measure#Concentration_on_the_sphere [10]: https://en.wikipedia.org/wiki/Talagrand%27s_concentration_inequality [11]: https://terrytao.wordpress.com/2010/01/03/254a-notes-1-concentration-of-measure/