Spectral Gerrymandering (2023)

Undergraduate: Andrew Sun

Faculty Advisor: Jeremy Marzuola
Department: Mathematics

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The literature around quantifying the extent to which a proposed districting map is a gerrymander has rapidly expanded in the last decade. Such a task is accomplished by sampling from an innumerable distribution of “reasonable” (often determined by the state constitution) maps, and then comparing the consortium of samples with the actual proposed map.

Sampling methodology has converged toward MCMC (Markov Chain Monte Carlo) methods, but researchers still know relatively little about the properties of different MCMC dynamics, including how well they mix, especially under perturbations to the parameter space.

In this project, we take a realistic enumerated example, the Duplin-Onslow county cluster in North Carolina, and utilize concepts from spectral graph theory to analyze the properties of two different dynamics established in the literature: Forest Metropolized Recombination and Reversible Recombination. We explore two remarkable properties with the later two algorithms: an extremely small spectral gap as gamma approaches 1, and a strong phase shift in the eigenvalue space. We then compare properties such as the invariant measure with another well-established dynamic, Regular Recombination, which does not enforce a known measure.

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