 Solution

Replace each XY model bond weight by its periodic Gaussian approximation. With angles integrated using $d\theta /(2\pi )$, the Villain model iswhere $N_b$ is the number of bonds. Changing the angle-measure normalization multiplies $Z_V$ by an explicit constant only. The Poisson summation formula applied to one bond gives the equivalent winding representationThus the same Villain model can be written $Z_V=\int {\prod }_{x}d{\theta }_x/(2\pi ){\sum }_{\{p_e\}}\exp [-\beta {\sum }_e({\Delta }_e\theta -2\pi p_e{)}^2/2]$. The coefficient $1/\sqrt{2\pi \beta }$ is the PDF's value; the converted TeX incorrectly places $\beta$ outside the square root. The cosine-to-Gaussian replacement is a large-$\beta$ approximation, but the rest of the calculation treats $Z_V$ as the exact model for every positive $\beta$.