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

Let $T_ε$ be the noise operator acting on Boolean functions $f:\{0, 1\}^n\to \{0, 1\}$, where $ε\in[0, 1/2]$ is the noise parameter. Given $α>1$ and fixed mean $\mathbb{E} f$, which Boolean function $f$ has the largest $α$-th moment $\mathbb{E}(T_εf)^α$? This question has close connections with noise stability of Boolean functions, the problem of non-interactive correlation distillation, and Courtade-Kumar's conjecture on the most informative Boolean function. In this paper, we characterize maximizers in some extremal settings, such as low noise ($ε=ε(n)$ is close to 0), high noise ($ε=ε(n)$ is close to 1/2), as well as when $α=α(n)$ is large. Analogous results are also established in more general contexts, such as Boolean functions defined on discrete torus $(\mathbb{Z}/p\mathbb{Z})^n$ and the problem of noise stability in a tree model.