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We develop several properties of the almost flat highest weights of$\mathrm{U}(N)$ and $\mathrm{SU}(N)$ introduced in a previous paper, and usethem to compute the limit of the expectation and variance of Wilson loopsassociated with contractible simple loops on compact orientable surfaces withgenus 1 and higher. As such, it provides a purely representation-theoreticproof for the main result of another paper by the author and A. Dahlqvist inthe case of unitary groups.

Almost flat highest weights and application to Wilson loops on compact surfaces