Brownian-bridge scaling for matchgate commutator graphs

This repository develops a paper proof of the Brownian-bridge area law for average distances in matchgate commutator-graph components. The main source is paper/main.tex. Exact finite-size expectation checks are implemented in code/exact_mean.cpp.

For κₙ/(2n)→ρ∈(0,1), the manuscript gives the distributional limit, the mean constant, and Var(D₂ₙ,κₙ)∼ρ(1−ρ)(28/15−π/2)n³. It also gives the exact central mean and its first n¹ᐟ² correction. The Haar-averaged graph-complexity consequence is stated separately from the Hamiltonian-dependent pointwise inequality between graph and Krylov complexity.

Build the manuscript with:

make paper

Build and run the exact expectation checker with:

make exact-mean
./build/exact_mean 100 200 400 800

The recorded table was generated with GCC 13.3.0 using C++20 and OpenMP. The program evaluates exact finite-sum identities with long double, lgammal, and normalized exponential weights; its decimal outputs are floating-point approximations rather than certified intervals.

Each command-line argument is the number of matchgate sites (n); the checker uses the central component (\kappa=n) unless --rho is supplied.

For the central component, the program evaluates the mean in two independent ways: the cumulative hypergeometric formula and the symmetric-difference bridge formula. It also reports the observed coefficient of the (\sqrt n) correction, whose theorem value is (-5\sqrt{2\pi}/64).

Recorded acceptance outputs are in results/exact_mean.tsv, and the source-level citation audit is in SOURCES.md.