The moduli space of flat connections on a surface is a fundamental object in geometry and physics. It parameterizes all the ways you can consistently assign group elements to paths on the surface. Its volume — computed by integrating over the entire space — encodes deep invariants of the surface and the group. But volume formulas derived from gauge theory and representation theory often involve alternating sums: positive and negative terms that partially cancel. The final answer is positive (it's a volume), but the formula isn't manifestly so. You have to trust the cancellation.
Francois et al. (arXiv: 2603.20517) derive a manifestly positive formula — one where every term is non-negative. The volume is expressed as a sum over polytopes describing “coloured honeycombs” on a polygon, inspired by Knutson and Tao's work on the eigenvalue problem for Hermitian matrices.
The connection to probability: the same formula arises as a marginal of the Yang-Mills measure, and the authors express it in terms of an explicit path process — a probabilistic description of how the connection evolves along paths on the surface.
The through-claim: the right formula reveals the right structure. The alternating-sum formulas compute the correct answer by delicate cancellation — the information about positivity is hidden in the algebra. The positive formula computes the same answer without cancellation — the positivity is manifest in the geometry. Both formulas are true, but only the positive one reflects the underlying structure: the volume is built from pieces (polytopes) that individually make geometric sense. The formula is not just a calculation; it's an explanation.
Francois, Garcia-Zelada, Levy & Tarrago, 2603.20517. Probability / gauge theory / moduli spaces / flat connections / manifestly positive formulas.