Non-interactive correlation distillation, inhomogeneous Markovchains, and the reverse Bonami-Beckner inequality

In this paper we study non-interactive correlation distillation (NICD), a generalization ofnoise sensitivity previously studied earlier. We extend the model to NICD on trees. In thismodel there is a fixed undirected tree with players at some of the nodes. One node is givena uniformly random string and this string is distributed throughout the network, with theedges of the tree acting as independent binary symmetric channels. The goal of the playersis 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 (the model considered earlier by Mossel and O\u27Donnell),we resolve the open question of whether the success probability must go to zero as $k \to \infty$.We show that this is indeed the case and provide matching upper and lower bounds on theasymptotically 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 touse the same 1-bit function.(3). In the general case we show that all players should use monotone functions. We also show,somewhat surprisingly, that 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. Although the usualBonami-Beckner has been frequently used before, its reverse counterpart seems very little-known;To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discretecube and a new result on the mixing of short random walks on the cube. Another tool that we needis a tight bound on the probability that a Markov chain stays inside certain sets; we prove a newtheorem generalizing and strengthening previous such bounds. On the probabilistic side, we usethe ``reflection principle\u27\u27 and the FKG and related inequalities in order to study the problem ongeneral trees