In this article, we prove irreducibility results for a family of Markov chains arising in the study of redistricting and detecting gerrymandering. These chains use ReCom moves as their transition mechanism and are commonly employed in Markov chain Monte Carlo methods to generate ensembles of districting plans. Such ensembles are frequently used for outlier analysis, in which a proposed districting map is compared against the ensemble to determine whether it behaves atypically; this methodology often appears in expert testimony in redistricting litigation. We show that when the underlying dual graph is a triangular subset of the triangular lattice and each district consists of two merged geographic regions, the associated ReCom chain is irreducible. This provides another entry in the very small list of known classes of ReCom chains for which irreducibility has been established. We also demonstrate the fragility of this phenomenon by constructing an infinite family of maps for which the corresponding ReCom chain is not irreducible. Indeed, we produce a districting map that, after implementing a single ReCom move, always yields the same original map. These examples remain structurally close to the triangular lattice: they arise as subdivisions of the triangular lattice, and the resulting graphs have maximum degree at most 8. Finally, we prove irreducibility for a further special case: the ReCom chain on a 3 x n grid graph partitioned into three districts of size n.

翻译：本文证明了一类出现在选区划分与选举操控检测研究中的马尔可夫链的不可约性结果。这些链以ReCom移动作为转移机制，在马尔可夫链蒙特卡洛方法中常用于生成选区划分方案的集合。此类方案集合频繁用于异常值分析，即通过将拟议的选区划分图与方案集合进行比较，判定其是否呈现异常行为——该方法论经常出现在选区划分诉讼中的专家证词中。我们证明：当底层对偶图为三角形晶格的三角子集，且每个选区由两个合并的地理区域组成时，相应的ReCom链是不可约的。这为极少数已知已建立不可约性的ReCom链类别增添了一个新实例。同时，通过构造无穷族映射，证明相应ReCom链不可约时该现象具有脆弱性——我们确实构造了一个选区划分图，在实施单次ReCom移动后始终恢复为原始映射。这些实例仍保持与三角形晶格的结构相似性：它们作为三角形格点的子分产生，所得图的最大度数不超过8。最后，我们证明另一种特殊情况的不可约性：划分为三个大小为n选区的3×n网格图上的ReCom链。