, Volume 154, Issue 1, pp 299-336

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

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Abstract

In this paper we studynon-interactive 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.

Our new contributions include the following:

  • • In the case of ak-leaf 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 slowly-decaying polynomial).

  • • In the case of thek-vertex path graph, we show that it is always optimal for all players to use the same 1-bit 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 Bonami-Beckner inequality. Although the usual Bonami-Beckner 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.

    Supported by a Miller fellowship in Statistics and CS, U.C. Berkeley, by an Alfred P. Sloan fellowship in Mathematics, and by NSF grant DMS-0504245.
    Most of this work was done while the author was a student at Massachusetts Institute of Technology. This material is based upon work supported by the National Science Foundation under agreement No. CCR-0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
    Most of this work was done while the author was at the Institute for Advanced Study, Princeton, NJ. Work supported by an Alon Fellowship, ARO grant DAAD19-03-1-0082 and NSF grant CCR-9987845.
    Supported in part by NSF grant DMS-0103841 the Swedish Research Council and the Göran Gustafsson Foundation (KVA).
    Research supported in part by NSF grant DMS-0106589, DMS-0355497, and by an Alfred P. Sloan fellowship.