  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.
