In three dimensions, the low-temperature Villain model's cosine-cosine correlation lies between c/|x|^2 and C(ln|x|)^{22}/|x|^2, matching the spin-wave prediction up to a logarithm; in all d at least 3 the upper bound is improved to (ln|x|)^k/|x|^{d-1}.
Quantitative stochastic homogenization and large-scale regularity , volume 352
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Parallel spin wave for the Villain model
In three dimensions, the low-temperature Villain model's cosine-cosine correlation lies between c/|x|^2 and C(ln|x|)^{22}/|x|^2, matching the spin-wave prediction up to a logarithm; in all d at least 3 the upper bound is improved to (ln|x|)^k/|x|^{d-1}.