Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels
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In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs et al. (in: 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS, IEEE Computer Society, 2006), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees.
KeywordsKesten–Stigum reconstruction bound Markov random fields on trees Distributional recursion Nonlinear dynamical system
Mathematics Subject Classification60K35 82B26 82B20
We give special thanks to the journal editor and two anonymous reviewers who provided us with many constructive and helpful comments. We also give special thanks to Sébastien Roch for his inspiring discussions and reading of some proofs in the first version of this paper. We truly appreciate the warm encouragements on finishing this complete version and helpful discussions on this topic as well as future extensions, from colleagues at the 2017 and 2018 Columbia-Princeton Probability Day, 2017 Northeast Probability Seminar, 2018 Frontier Probability Days, 2017 and 2018 Finger Lakes Probability Seminar, and 2017 and 2018 Seminar on Stochastic Processes.
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