Theorem. The matched Wilson and Villain laws differ [fdip-007E]
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  • November 2, 2026

A single \(U(1)\) plaquette has the normalized Wilson or Villain weight. Matching their fundamental Fourier coefficients determines a Villain coupling for every Wilson coupling. The two probability laws still differ.

For \(\beta >0\), write \(r(\beta )=I_1(\beta )/I_0(\beta )\) and \(\beta _V(\beta )=-\frac {1}{2\log r(\beta )}\). Then \(0<r(\beta )<1\), the matched Villain coefficient is \(r(\beta )^{n^2}\), and \[\frac {\widehat c_n^W}{\widehat c_n^V}\to \infty \] as \(|n|\to \infty \). In particular the two normalized weights are distinct, so they define distinct probability laws on the circle. The second Wilson coefficient is \(1-\frac {2r(\beta )}{\beta }\).

Proof. 

  • November 2, 2026

The series for \(I_1\) has positive terms, so \(r>0\). With normalized Haar measure, \[\int (1-\cos \theta )\,e^{\beta \cos \theta }\,\frac {d\theta }{2\pi } =I_0(\beta )-I_1(\beta ).\] The integrand is nonnegative and not identically zero, and the exponential is positive, so the integral is positive. Thus \(r<1\), the logarithm is negative, and \(\beta _V\) is positive. The Villain normalization in the coefficient note gives \(\widehat c_n^V=\exp (-n^2/(2\beta _V))\). Substituting the displayed \(\beta _V\) produces \(1/(2\beta _V)=-\log r\), hence \(\widehat c_n^V=r^{n^2}\).

The \(j=0\) term of the series for \(I_n(\beta )\), with \(n\ge 0\), is \((\beta/2)^n/n!\). Every term is positive, so \(I_n(\beta )\ge (\beta/2)^n/n!\). Therefore \(\widehat c_n^W=I_n(\beta )/I_0(\beta )\) satisfies \[\frac {\widehat c_n^W}{r^{n^2}} \ge \frac {(\beta/2)^n}{n!\,I_0(\beta )\,r^{n^2}}.\] For \(n\ge 1\), one has \(n!\le n^n\), so the logarithm of the right-hand side is at least \(n\log (\beta/2)-n\log n-\log I_0(\beta )-n^2\log r\). The last summand tends to \(+\infty \) because \(\log r<0\), and it dominates the remaining terms. The lower bound tends to infinity, and so does the ratio, as \(n\to +\infty \). The series in the coefficient note depends on \(|n|\), so \(I_n(\beta )=I_{-n}(\beta )\). The matched Villain coefficient \(r^{n^2}\) is likewise unchanged when \(n\) is replaced by \(-n\). The ratio at \(-n\) equals the ratio at \(n\), and the same limit holds as \(|n|\to \infty \).

The coefficient note identifies each normalized continuous weight with its Fourier series. Some coefficient of large order differs, so the weights differ as functions and as probability laws.

The same series gives the second coefficient. The identity \(\frac {2}{\beta }(\beta/2)^{2j+1}=(\beta/2)^{2j}\) rewrites \[\frac {2}{\beta }I_1(\beta ) =\sum _{j\ge 0}\frac {(\beta/2)^{2j}}{j!\,(j+1)!}.\] Subtracting from \(I_0\) cancels the \(j=0\) term. For \(j\ge 1\) the difference of the summands is \(\frac {(\beta/2)^{2j}\,j}{j!\,(j+1)!} =\frac {(\beta/2)^{2j}}{(j-1)!(j+1)!}\), which is the \(j-1\) term of \(I_2\). Hence \(I_0-I_2=\frac {2}{\beta }I_1\), and \(\widehat c_2^W=1-\frac {2r(\beta )}{\beta }\). At \(\beta =2\) this coefficient is \(1-r(2)\), while the matched Villain coefficient is \(r(2)^4\).

The comparison is a one-plaquette identity. It does not assert a many-plaquette continuum limit.