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

Journal article, 2006

In this paper we study non-interactive correlation distillation (NICD), a generalization of noise sensitivity previously studied earlier. 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 (the model considered earlier by Mossel and O'Donnell), 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 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. 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 usual Bonami-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 discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds. On the probabilistic side, we use the ``reflection principle'' and the FKG and related inequalities in order to study the problem on general trees.