Exact spectral gap of the down-up walk on matroids with parallel elements

This deposit contains the preprint "Exact spectral gap of the down-up walk on matroids with parallel elements" together with its full replication package. The paper. For a matroid of rank r, the down-up (basis-exchange) walk on its bases has spectral gap at least 1/r, by the log-concavity theorem of Anari, Liu, Oveis Gharan and Vinzant (STOC 2019), and at most 2/r under a mild condition, by a conductance argument of Alev and Lau (STOC 2020). This note determines when the lower end of that window is attained by a degree-1 eigenfunction: a linear observable is an eigenfunction of eigenvalue exactly 1 − 1/r if and only if an explicit functional is constant on bases, and any pair of parallel elements makes it so. Consequently, every matroid with two parallel elements has down-up gap exactly 1/r; for the cographic matroid of a graph, this covers every graph with two edges in series — hence every graph with a degree-two vertex, which includes essentially every sparse graph arising in applications. At the other end of the window, the uniform matroids U(r, 2r−2) are shown to attain the 2/r ceiling exactly, so both ends of the interval [1/r, 2/r] are realized by explicit matroid families. The chain-dependence of these statements is treated with care: four distinct Markov chains from the literature are tabulated on a single running example (the doubled path of Montenegro and Son, STOC 2001, whose mechanism the main theorem explains and generalizes), where their spectral gaps differ by a factor Θ(m). The replication package (ZIP archive) contains the code, seeds, and recorded outputs behind every numerical claim in the paper: exact rational arithmetic verification of the four-chain table, exact enumeration and spectral computation for the control table (including the K₄ edge-subdivision experiment isolating the mechanism), and the kernel computation of the eigenfunction spaces, with an automated test suite and a README mapping each claim of the paper's verification section to the script that produces it.