In the study of Ising models on large locally tree-like graphs, in both rigorous and non-rigorous methods one is often led to understanding the so-called belief propagation distributional recursions and its fixed points. We prove that there is at most one non-trivial fixed point for Ising models with zero or certain random external fields. Previously this was only known for sufficiently ``low-temperature'' models. Our main innovation is in applying information-theoretic ideas of channel comparison leading to a new metric (degradation index) between binary-input-symmetric (BMS) channels under which the Belief Propagation (BP) operator is a strict contraction (albeit non-multiplicative). A key ingredient of our proof is a strengthening of the classical stringy tree lemma of (Evans-Kenyon-Peres-Schulman'00). Our result simultaneously closes the following 6 conjectures in the literature: 1) independence of robust reconstruction accuracy to leaf noise in broadcasting on trees (Mossel-Neeman-Sly'16); 2) uselessness of global information for a labeled 2-community stochastic block model, or 2-SBM (Kanade-Mossel-Schramm'16); 3) optimality of local algorithms for 2-SBM under noisy side information (Mossel-Xu'16); 4) uniqueness of BP fixed point in broadcasting on trees in the Gaussian (large degree) limit (ibid); 5) boundary irrelevance in broadcasting on trees (Abbe-Cornacchia-Gu-Polyanskiy'21); 6) characterization of entropy (and mutual information) of community labels given the graph in 2-SBM (ibid).

翻译：在大规模局部树状图上的伊辛模型研究中，无论是严谨方法还是非严谨方法，通常需要理解所谓的置信传播分布递归及其不动点。我们证明，在零外部场或特定随机外部场条件下，伊辛模型至多存在一个非平凡不动点。此前该结论仅对充分"低温"模型成立。我们的主要创新在于应用信道比较的信息论思想，提出一种新的度量（退化指数），使得在二进制输入对称（BMS）信道下，置信传播（BP）算子成为严格压缩映射（虽非乘法形式）。证明的关键要素是对经典弦树引理（Evans-Kenyon-Peres-Schulman'00）的强化。我们的结果同时解决了文献中的以下六大猜想：1）树状广播中鲁棒重建精度对叶子噪声的独立性（Mossel-Neeman-Sly'16）；2）标记二社区随机分块模型（2-SBM）中全局信息的无用性（Kanade-Mossel-Schramm'16）；3）含噪侧信息下2-SBM局部算法的最优性（Mossel-Xu'16）；4）高斯（大度数）极限下树状广播中BP不动点的唯一性（同上）；5）树状广播中边界无关性（Abbe-Cornacchia-Gu-Polyanskiy'21）；6）给定图结构下2-SBM社区标签熵（及互信息）的刻画（同上）。