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arXiv · 2005.07801

Broadcasting on trees near criticality

Yuzhou Gu·Hajir Roozbehani·Yury Polyanskiy
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Abstract

We revisit the problem of broadcasting on $d$-ary trees: starting from a Bernoulli$(1/2)$ random variable $X_0$ at a root vertex, each vertex forwards its value across binary symmetric channels $\mathrm{BSC}_\delta$ to $d$ descendants. The goal is to reconstruct $X_0$ given the vector $X_{L_h}$ of values of all variables at depth $h$. It is well known that reconstruction (better than a random guess) is possible as $h\to \infty$ if and only if $\delta < \delta_c(d)$. In this paper, we study the behavior of the mutual information and the probability of error when $\delta$ is slightly subcritical. The innovation of our work is application of the recently introduced "less-noisy" channel comparison techniques. For example, we are able to derive the positive part of the phase transition (reconstructability when $\delta<\delta_c$) using purely information-theoretic ideas. This is in contrast with previous derivations, which explicitly analyze distribution of the Hamming weight of $X_{L_h}$ (a so-called Kesten-Stigum bound).

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Yuzhou Gu, Hajir Roozbehani, Yury Polyanskiy. 2020-05-15. Broadcasting on trees near criticality. https://arxiv.org/abs/2005.07801

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