Journal of Statistical Physics

, Volume 177, Issue 3, pp 438–467 | Cite as

Information Reconstruction on an Infinite Tree for a \(4\times 4\)-State Asymmetric Model with Community Effects

  • Wenjian Liu
  • Ning NingEmail author


The reconstruction problem on an infinite tree, is to collect and analyze massive data samples at the nth level of the tree to identify whether there is non-vanishing information of the root, as n goes to infinity. This problem has wide applications in various fields such as biology, information theory, and statistical physics. Although it has been studied in numerous contexts, the existing literature with rigorous reconstruction thresholds established are very limited. In this paper, we study the noise channel in terms of a \(4\times 4\)-state asymmetric transition matrix with community effects. By means of refined analyses of moment recursion, in-depth concentration estimates, and thorough investigations on an asymptotic 4-dimensional nonlinear second order dynamical system, we give the first rigorous result on asymmetric noisy channels with community effects by providing the exact conditions that the reconstruction bound is not tight, which yields the hybrid-hard phase in corresponding stochastic block models.


Kesten–Stigum reconstruction bound Markov random fields on trees Distributional recursion Nonlinear dynamical system 

Mathematics Subject Classification

60K35 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 Joseph Felsenstein and Sébastien Roch for conversations on this subject.


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Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Queensborough Community CollegeCity University of New YorkNew YorkUSA
  2. 2.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

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