在临界附近的树上广播
Broadcasting on trees near criticality
摘要：
我们重新审视$d$ -ary树上的广播问题：从根顶点处的Bernoulli $(1/2)$随机变量$X_0$开始，每个顶点将其值通过二进制对称通道$\mathrm{BSC}_δ$转发到$d$后代。目标是在给定深度$h$处的所有变量的值的向量$X_{L_h}$的情况下重建$X_0$。众所周知，当且仅当$δ< δ_c(d)$时，重建（比随机猜测更好）可能作为$h\to \infty$。本文研究了当$δ$为弱次临界时互信息的行为和错误概率。我们工作的创新之处是应用了最近推出的“低噪声”信道比较技术。例如，我们能够使用纯粹的信息论思想导出相变的正部分（当$δ<δ_c$时的可重构性）。这与以前的推导相反，以前的推导明确地分析了$X_{L_h}$的汉明重量的分布（所谓的Kesten-Stigum界限）。
英文摘要：
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 $\mathrm{BSC}_δ$ 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 $δ< δ_c(d)$. In this paper, we study the behavior of the mutual information and the probability of error when $δ$ 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 $δ<δ_c$) 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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
