The local behavior of typical solutions of random constraint satisfaction problems (CSP) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, Muller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular NAE-SAT model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters, a result rigorously verified by Nam, Sly, Sohn (2021). This is the first characterization of the local weak limit in the condensation regime for any sparse random CSPs in the so-called one-step replica symmetry breaking (1RSB) class. Our result is non-asymptotic, and characterizes the tight fluctuation $O(n^{-1/2})$ around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1RSB belief-propagation fixed point. We believe that our proof technique has broad applicability to random CSPs in the 1RSB class.

翻译：随机约束满足问题（CSP）典型解的局部行为描述了众多重要现象，包括聚类阈值、相关性衰减以及消息传递算法的行为。当约束密度较低时，研究植入模型成为确定这种局部行为的有效技术，其典型结构具有简单的马尔可夫性。Coja-Oghlan、Kapetanopoulos、Muller（2020）的研究表明，对于一大类模型，该描述适用于所谓的凝聚阈值之前。凝聚阈值后的局部行为因长程相关性而更为复杂。本文重新研究了凝聚态下的随机正则NAE-SAT模型，确定了描述典型变量周围随机解的局部弱极限。该极限呈现出复杂的非马尔可夫结构，源于解空间由少数大团簇主导，这一结论由Nam、Sly、Sohn（2021）严格证实。这是首次对所谓一步副本对称破缺（1RSB）类稀疏随机CSP在凝聚态下的局部弱极限进行刻画。我们的结果是非渐近的，并描述了极限周围的紧波动$O(n^{-1/2})$。证明基于将编码团簇结构的无穷自旋系统的局部邻域与树上的广播模型耦合，该广播模型的信道由1RSB置信传播不动点给出。我们相信该证明技术对1RSB类随机CSP具有广泛适用性。