Non-interactive correlation distillation, inhomogeneous Markov chains,   and the reverse Bonami-Beckner inequality
  In this paper we study non-interactive correlation distillation (NICD), a generalization of the study of noise sensitivity of boolean functions. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: 1. In the case of a k-leaf star graph, we resolve the open question of whether the success probability must go to zero as k goes to infinity. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial). 2. In the case of the k-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function. 3. In the general case we show that all players should use monotone functions. 4. For certain trees it is better if not all players use the same function. Our techniques include the use of the reverse Bonami-Beckner inequality. 
Elchanan Mossel, Ryan O'Donnell, Oded Regev, Jeffrey Steif, Benjamin   Sudakov