Example. A self-loop with nonzero expectation [fdip-006T]

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For a self-loop \(e\), the transformation is only conjugation. In the square-cell convention, take one vertex, one loop and one square whose attaching walk traverses that loop 4 times. Use the finite Wilson law. With \(G=\{1,\omega ,\omega ^2\}\) and \(\omega =e^{2\pi i/3}\), use the one-dimensional representation \(\rho (U)=U\). Then \(U_p=U^4=U\), the gauge action is trivial, and \[\langle U\rangle _W =\frac {1-e^{-3\beta/2}}{1+2e^{-3\beta/2}}>0\qquad (\beta >0).\] Thus no general open-edge vanishing conclusion extends to self-loops.