Proposition. Wilson and Villain coefficients on the circle [fdip-007D]
✍️source

  • November 1, 2026

For \(G=U(1)\), use characters \(\chi _n(e^{i\theta })=e^{in\theta }\) with \(n\in \mathbb Z\) and normalized Haar measure \(d\theta /(2\pi )\). For \(\beta >0\), the Wilson weight \(w_\beta (\theta )=e^{\beta \cos \theta }\) has coefficients \[I_n(\beta )=\sum _{j=0}^\infty \frac {(\beta /2)^{2j+|n|}}{j!(j+|n|)!}>0, \qquad \sum _{n\in \mathbb Z}I_n(\beta )=e^\beta .\] Thus its character series is uniformly absolutely convergent. The Wilson action convention includes the additional constant factor \(e^{-\beta }\); this multiplies all coefficients but cancels from the normalized one-plaquette probability.

TauCeti's Real.integral_exp_mul_cos_mul_cexp proves the Wilson circle integral with exponent \(+in\theta \) and unnormalized measure on \([-\pi ,\pi ]\). Replacing \(n\) by \(-n\) and dividing by \(2\pi \) gives the coefficient convention here; its Bessel value has the displayed power series. This correspondence supports the Wilson coefficient identity, not the Villain formula or the comparison of the two weights.

For \(\beta _V>0\), the Villain weight is the periodic Gaussian \[v_{\beta _V}(\theta )=\sum _{\ell \in \mathbb Z} \exp \left [-\frac {\beta _V}{2}(\theta -2\pi \ell )^2\right ].\] Its character coefficient is \[c_n^V=(2\pi \beta _V)^{-1/2} e^{-n^2/(2\beta _V)}.\] The coefficient series is again uniformly absolutely convergent and all coefficients are strictly positive. Dividing each weight by its zeroth coefficient normalizes its integral to one, giving \[\widehat c_n^W=\frac {I_n(\beta )}{I_0(\beta )},\qquad \widehat c_n^V=e^{-n^2/(2\beta _V)}.\] These normalized coefficients differ from the unnormalized \(c_\rho \) in the surface formula.

Proof. 

  • November 1, 2026

Put \(z=e^{i\theta }\). Multiply the two absolutely convergent series for \(\exp (\beta z/2)\) and \(\exp (\beta z^{-1}/2)\). The coefficient of \(z^n\) is the displayed \(I_n\); the total of all absolute coefficients is the product \(e^{\beta /2}e^{\beta /2}\). Every \(I_n\) is positive for \(\beta >0\).

For the periodic Gaussian, uniform Gaussian tail bounds on a period justify continuity and integration term by term. Unfolding its coefficient integral gives \[c_n^V=\frac 1{2\pi }\int _{\mathbb R} e^{-\beta _V x^2/2}e^{-inx}\,dx.\] For real \(t\), let \(J(t)\) be the integral without \(1/(2\pi )\) and with \(n\) replaced by \(t\). Gaussian domination permits differentiation under the integral. Integration by parts, with vanishing boundary terms, gives \(J'(t)=-(t/\beta _V)J(t)\). Since \(J(0)=\sqrt {2\pi /\beta _V}\), this proves the coefficient formula. The Gaussian coefficient bound is summable over \(n\). Its uniformly convergent Fourier series has the same coefficients as the continuous periodic Gaussian, so completeness of the character basis and continuity identify the two functions. Finally, the integral of a weight is its zeroth coefficient, which proves the normalization.

Both positive-parameter weights satisfy the positive-character kernel hypotheses: they are real, strictly positive, invariant under inversion, and have positive summable coefficients. For \(U(1)\) every irreducible dimension is one. This checks the kernel assumptions without asserting any transfer-operator or continuum conclusion.