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 correlation decay conditions. More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs with correlation decay. We show how some of the original applications of the theory of in terms of voting and coalitions extend to general measures with correlation decay. 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") with correlation decay.

翻译：乘积测度中的影响力理论在理论计算机科学、组合数学和离散概率领域具有深远应用。该深刻理论与泛函不等式及离散群上的傅里叶分析密切相关。最初，函数影响力的概念源于社会选择理论——在该理论中，布尔函数代表投票方案，其输入对应选票，输出表示选举结果。因此，乘积测度描述了一种各方选票随机独立分布的场景，但这往往与现实情境相去甚远。本文开始为满足混合或相关性衰减条件的更一般测度发展影响力理论。具体而言，我们针对具有相关性衰减的有界度图上的马尔可夫随机场，证明了KKL和Talagrand影响力定理的对应版本。我们展示了原理论在投票与联盟方面的部分应用如何推广到具有相关性衰减的一般测度。我们的结果既揭示了关联选民投票行为的规律，也阐明了具有相关性衰减的马尔可夫随机场（亦称"自旋系统"）上一般函数的特性。