SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1381-1417, January 2021.
In this paper, we analyze the tree reconstruction problem, which is to determine whether symbols at the $n$th level of the tree provide information on the root as $n\rightarrow \infty$. This problem has wide applications in various fields such as biology, information theory, and statistical physics, and its close connections to the clustering problem in the setting of the stochastic block model (SBM) have been well established. For SBM, an “information-theoretically-solvable-but-computationally-hard" region, or say “hybrid-hard" phase, appears whenever the reconstruction bound of the corresponding reconstruction on the tree problem is not tight. Inspired by the recently proposed $q_1+q_2$ SBM [F. Ricci-Tersenghi, G. Semerjian, and L. Zdeborová, Phys. Rev. E, 99 (2019), 042109], we extend the classical works on the Ising model [C. Borgs et al., Proceedings of the 47th Annual IEEE Symposium on Foundation of Computer Science, 2006, pp. 518--530] and the Potts model [A. Sly, Ann. Probab., 39 (2011), pp. 1365--1406] by studying a general model which incorporates the characteristics of both Ising and Potts through different in-community and out-community transition probabilities and rigorously give the conditions for the nontightness of the reconstruction threshold.

中文翻译：

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无限树上具有不同社区内和社区外突变的一般模型的可重构性的相变

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1381-1417 页，2021 年 1 月。
在本文中，我们分析了树重建问题，即确定树第 $n$ 层的符号是否提供关于根的信息为 $n\rightarrow\infty$。这个问题在生物学、信息论和统计物理学等各个领域都有广泛的应用，并且它与随机块模型（SBM）设置中的聚类问题的密切联系已经很好地建立起来。对于 SBM，每当树问题上相应重建的重建界限不严格时，就会出现“信息理论上可解决但计算困难”的区域，或者说“混合困难”阶段。受最近提出的 $q_1+q_2$ SBM [F. Ricci-Tersenghi、G. Semerjian 和 L. Zdeborová，物理学。Rev. E, 99 (2019), 042109]，我们扩展了 Ising 模型上的经典著作 [C. Borgs 等人，第 47 届 IEEE 计算机科学基础研讨会论文集，2006 年，第 518--530 页]和 Potts 模型 [A. 狡猾，安。Probab., 39 (2011), pp. 1365--1406] 通过研究一个通用模型，该模型通过不同的社区内和社区外转移概率结合了 Ising 和 Potts 的特征，并严格给出了非紧密性的条件重构阈值。