Noninteractive correlation distillation, inhomogeneous Markov chains, and the reverse BonamiBeckner inequality
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Abstract
In this paper we studynoninteractive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD 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.

• In the case of akleaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero ask » ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowlydecaying polynomial).

• In the case of thekvertex path graph, we show that it is always optimal for all players to use the same 1bit function.

• 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 thereverse BonamiBeckner inequality. Although the usual BonamiBeckner has been frequently used before, its reverse counterpart seems not to be well 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 [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.
Keywords
Markov Chain Boolean Function Isoperimetric Inequality Optimal Protocol Random StringPreview
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