🤖 AI Summary
This study investigates the irreducibility of ReCom-based Markov chains in political redistricting to ensure representative sampling of districting plans. Employing tools from graph theory and Markov chain Monte Carlo methods, the authors establish—for the first time—the irreducibility of the two-partition ReCom chain on subgraphs of the triangular lattice and prove irreducibility for three-equal-partition ReCom chains on 3×n grid graphs. Conversely, they construct an infinite family of counterexamples demonstrating that even minor structural variations in closely related planar subdivisions can render the chain disconnected, highlighting the fragility of irreducibility. These results delineate critical theoretical boundaries for assessing fairness in redistricting algorithms.
📝 Abstract
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.