Sampling-based methods such as ReCom are widely used to audit redistricting plans for fairness, with the balanced spanning tree distribution playing a central role since it favors compact, contiguous, and population-balanced districts. However, whether such samples are truly representative or exhibit hidden biases remains an open question. In this work, we introduce the notion of separation fairness, which asks whether adjacent geographic units are separated with at most a constant probability (bounded away from one) in sampled redistricting plans. Focusing on grid graphs and two-district partitions, we prove that a smooth variant of the balanced spanning tree distribution satisfies separation fairness. Our results also provide theoretical support for popular MCMC methods like ReCom, suggesting that they maintain fairness at a granular level in the sampling process. Along the way, we develop tools for analyzing loop-erased random walks and partitions that may be of independent interest.

翻译：基于抽样的方法（如ReCom）被广泛用于审计选区重划方案的公平性，其中平衡生成树分布因其倾向于紧凑、连续且人口平衡的选区而发挥着核心作用。然而，此类样本是否真正具有代表性或存在隐藏偏差仍是一个悬而未决的问题。本文引入了分离公平性的概念，该概念关注在抽样得到的选区重划方案中，相邻地理单元被划分到不同选区的概率是否不超过一个常数（严格小于1）。聚焦于网格图和双分区情形，我们证明了平衡生成树分布的一个光滑变体满足分离公平性。我们的研究结果也为ReCom等流行的马尔可夫链蒙特卡洛方法提供了理论支持，表明它们在抽样过程中能在微观层面保持公平性。在此过程中，我们发展了一套用于分析循环擦除随机游走与划分的工具，这些工具可能具有独立的学术价值。