Finitary Boolean functions

Abstract: We study functions on the infinite-dimensional Hamming cube ${-1,1}^\infty$, in particular Boolean functions into ${-1,1}$, generalising results on analysis of Boolean functions on ${-1,1}^n$ for $n\in\mathbb{N}$. The notion of noise sensitivity, first studied in arXiv:math/9811157 , is extended to this setting, and basic Fourier formulas are established. We also prove hypercontractivity estimates for these functions, and give a version of the Kahn-Kalai-Linial theorem giving a bound relating the total influence to the maximal influence. Particular attention is paid to so-called finitary functions, which are functions for which there exists an algorithm that almost surely queries only finitely many bits. Two versions of the Benjamini-Kalai-Schramm theorem characterizing noise sensitivity in terms of the sum of squared influences are given, under additional moment hypotheses on the amount of bits looked at by an algorithm. A version of the Kahn-Kalai-Linial theorem giving that the maximal influence is of order $\frac{\log(n)}{n}$ is also given, replacing $n$ with the expected number of bits looked at by an algorithm. Finally, we show that the result in arXiv:math/0504586 that revealments going to zero implies noise sensitivity also holds for finitary functions, and apply this to show noise sensitivity of a version of the voter model on sufficiently sparse graphs.