Motivated by the theory of spin-glasses in physics, we study the so-called reconstruction problem for the related distributions on the tree, and on the sparse random graph $G(n,d/n)$. Both cases, reduce naturally to studying broadcasting models on the tree, where each edge has its own broadcasting matrix, and this matrix is drawn independently from a predefined distribution. In this context, we study the effect of the configuration at the root to that of the vertices at distance $h$, as $h\to\infty$. We establish the reconstruction threshold for the cases where the broadcasting matrices give rise to symmetric, 2-spin Gibbs distributions. This threshold seems to be a natural extension of the well-known Kesten-Stigum bound which arises in the classic version of the reconstruction problem. Our results imply, as a special case, the reconstruction threshold for the well-known Edward-Anderson model of spin-glasses on the tree. Also, we extend our analysis to the setting of the Galton-Watson tree, and the random graph $G(n,d/n)$, where we establish the corresponding thresholds.Interestingly, for the Edward-Anderson model on the random graph, we show that the replica symmetry breaking phase transition, established in [Guerra and Toninelli:2004], coincides with the reconstruction threshold. Compared to the classical Gibbs distributions, the spin-glasses have a lot of unique features. In that respect, their study calls for new ideas, e.g., here we introducing novel estimators for the reconstruction problem. Furthermore, note that the main technical challenge in the analysis is the presence of (too) many levels of randomness. We manage to circumvent this problem by utilising recently proposed tools coming from the analysis of Markov chains.

翻译：受物理学中自旋玻璃理论的启发，我们研究了树上及相关分布（包括稀疏随机图$G(n,d/n)$）中所谓的重构问题。这两种情形都自然地简化为树上的广播模型，其中每条边具有各自的广播矩阵，且该矩阵独立地从预定义分布中抽取。在此背景下，我们研究了当$h\to\infty$时，根节点构型对距离$h$处顶点构型的影响。我们为广播矩阵产生对称二自旋吉布斯分布的情形建立了重构阈值。该阈值似乎是经典重构问题中著名的Kesten-Stigum界的自然推广。作为特例，我们的结果给出了树上著名的爱德华-安德森自旋玻璃模型的重构阈值。此外，我们将分析扩展到高尔顿-沃森树和随机图$G(n,d/n)$的设定，并建立了相应的阈值。有趣的是，对于随机图上的爱德华-安德森模型，我们证明[Guerra and Toninelli:2004]中建立的副本对称破缺相变与重构阈值一致。与经典吉布斯分布相比，自旋玻璃具有许多独特特征。因此，对其研究需要新的思路，例如本文为重构问题引入了新颖的估计量。此外，需注意分析中的主要技术挑战在于存在（过多）层次的随机性。我们通过利用近期提出的马尔可夫链分析工具成功绕过了这一问题。