TY - GEN

T1 - Influences in Mixing Measures

AU - Koehler, Frederic

AU - Lifshitz, Noam

AU - Minzer, Dor

AU - Mossel, Elchanan

N1 - Publisher Copyright: © 2024 Copyright is held by the owner/author(s). Publication rights licensed to ACM.

PY - 2024/6/10

Y1 - 2024/6/10

N2 - 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.

AB - 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.

KW - Analysis of Boolean Functions

KW - Influences of Variables

KW - Non-product Measures

UR - http://www.scopus.com/inward/record.url?scp=85196632363&partnerID=8YFLogxK

U2 - https://doi.org/10.1145/3618260.3649731

DO - https://doi.org/10.1145/3618260.3649731

M3 - منشور من مؤتمر

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 527

EP - 536

BT - STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing

A2 - Mohar, Bojan

A2 - Shinkar, Igor

A2 - O�Donnell, Ryan

T2 - 56th Annual ACM Symposium on Theory of Computing, STOC 2024

Y2 - 24 June 2024 through 28 June 2024

ER -