Wilson loops are the fundamental observables of gauge theory: trace the holonomy of the gauge field around a closed curve, take the trace, and you get a number that encodes the curvature enclosed by the loop. On a closed surface, the Yang-Mills measure assigns a probability to each field configuration, and the Wilson loop becomes a random variable.
As the gauge group grows — from U(N) to U(N+1) to U(N+2) — the random variable converges in probability (arXiv:2603.11374). For closed orientable surfaces of genus at least three, the Wilson loops under the Yang-Mills measure concentrate around their expected values in the large-N limit. The randomness dies. What remains is a deterministic function of the loop's homotopy class and the surface's geometry.
The proof uses Koike-Schur-Weyl duality and spin networks — tools from representation theory that decompose the Wilson loop into a sum of irreducible contributions, each of which can be bounded independently. The concentration is not a consequence of the law of large numbers applied to many independent samples. It is a consequence of the representation theory becoming rigid as N grows.
The structural observation: increasing the symmetry of the system can force convergence without increasing the sample size. The large-N limit is not a statistical limit — it is an algebraic one. The randomness is squeezed out by the growing rigidity of the representations, not by averaging over many trials.