Ensemble analysis has become an important tool for analyzing and quantifying gerrymandering; the main idea is to generate a large, random sample of districting plans (an "ensemble") to which any proposed plan may be compared. If a proposed plan is an extreme outlier compared to the ensemble with regard to various redistricting criteria, this may indicate that the plan was deliberately engineered to produce a specific outcome. A variety of methods have been used to construct ensembles of plans, and a fundamental question that arises is: How accurately does an ensemble constructed by a particular method represent the entire space of valid plans -- or, if a method has an inherent bias towards particular types of plans, can this bias be identified and quantified? Recently, Markov Chain Monte Carlo (MCMC) methods have become a predominant tool for constructing ensembles of plans. In this paper, we focus on the MCMC method known as "ReCom," which was introduced in 2018 by the Metric Geometry and Gerrymandering Group. This method appears to produce plans with relatively compact districts compared to some other methods, and we sought to understand this phenomenon in greater detail. In order to model the basic ReCom step, we constructed large ensembles of plans consisting of two districts for two grid graphs and for the precinct graph of Boulder County, CO. We found that, to a high degree of accuracy, the sampling probability for any particular plan is proportional to an exponentially decaying function of a discrete measure that approximates the length of the boundary between the two districts in the plan. This suggests a more quantitative formulation of the observation that ReCom tends to produce relatively compact districts, and it represents an important first step towards understanding the full sampling probability distribution associated to the ReCom method.

中文翻译：

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在ReCom抽样方法中量化区域紧凑度的第一步

合奏分析已成为分析和量化搬运需求的重要工具。主要思想是生成一个大型的，随机的选区计划样本（“整体”），可以将任何拟议的计划与该计划进行比较。如果就各种重新划分标准而言，拟议的计划与总体相比是一个极端的异常值，则这可能表明该计划是经过精心设计的，可以产生特定的结果。已经使用了多种方法来构建计划集合，并且出现了一个基本问题：用特定方法构建的集合如何准确地表示有效计划的整个空间-或者，如果一种方法固有地偏向于特定类型的计划，是否可以识别和量化这种偏见？最近，马尔可夫链蒙特卡罗（MCMC）方法已成为构建整体计划的主要工具。在本文中，我们重点介绍由Metric Geometry and Gerrymandering Group在2018年引入的称为“ ReCom”的MCMC方法。与其他方法相比，此方法似乎产生的区域相对紧凑的计划，并且我们试图更详细地了解这种现象。为了对基本的ReCom步骤建模，我们为两个网格图和科罗拉多州博尔德县的区域图构建了由两个区域组成的大型计划集合。我们发现，从高度上讲，任何区域的抽样概率特定计划与离散量的指数衰减函数成比例，该离散量近似于计划中两个区域之间的边界长度。