Influences in Mixing Measures

Frederic Koehler, Noam Lifshitz, Dor Minzer, Elchanan Mossel

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability. This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups. Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes, and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently distributed, which is often far from the truth in real-life scenarios. We begin to develop the theory of influences for more general measures under mixing or spectral independence conditions. More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs when the Glauber dynamics mix rapidly. We thus resolve a long standing challenge, stated for example by Kalai and Safra (2005). We show how some of the original applications of the theory of in terms of voting and coalitions extend to these general dependent measures. Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called "spin-systems") where the Glauber dynamics mixes rapidly.

Original languageEnglish
Title of host publicationSTOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
EditorsBojan Mohar, Igor Shinkar, Ryan O�Donnell
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Electronic)9798400703836
StatePublished - 10 Jun 2024
Externally publishedYes
Event56th Annual ACM Symposium on Theory of Computing, STOC 2024 - Vancouver, Canada
Duration: 24 Jun 202428 Jun 2024

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference56th Annual ACM Symposium on Theory of Computing, STOC 2024

Bibliographical note

Publisher Copyright:
© 2024 Copyright is held by the owner/author(s). Publication rights licensed to ACM.


  • Analysis of Boolean Functions
  • Influences of Variables
  • Non-product Measures


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