Boolean functions: noise stability, non-interactive correlation, and mutual information

Let ǫ ∈ [0, 1/2] be the noise parameter and p > 1. We study the isoperimetric problem that for fixed mean Ef which Boolean function f maximizes the p-th moment E(Tǫf) p of the noise operator Tǫ acting on Boolean functions f : {0, 1} n 7→ {0, 1}. Our findings are: in the low noise scenario, i.e., ǫ is small, the maximum is achieved by the lexicographical function; in the high noise scenario, i.e., ǫ is close to 1/2, the maximum is achieved by Boolean functions with the maximal degree-1 Fourier weight; and when p is a large integer, the maximum is achieved by some monotone function, which particularly implies that, among balanced Boolean functions, the maximum is achieved by any function which is 0 on all strings with fewer than n/2 1’s. Our results recover Mossel and O’Donnell’s results about the problem of non-interactive correlation distillation, and confirm Courtade and Kumar’s Conjecture on the most informative Boolean function in the low noise and high noise regimes. We also observe that Courtade and Kumar’s Conjecture is equivalent to that the dictator function maximizes E(Tǫf) p for p close to 1.