Critical two-point decay $r^{-1/4}(\log r)^{1/8}$ in the planar XY model

Log-correction paper, Theorem 1.1: $C_{b_c}(r)=B_{XY}r^{-1/4}(\log r)^{1/8}(1+o(1))$ as $r\to\infty$ along the axis, with $B_{XY}\in(0,\infty)$, for the free-box limit. Exponent paper (September 24): $C_{b_c}(r)=r^{-1/4+o(1)}$. The constant is not computed; off-axis directions, other lattices and the Villain model at criticality are not covered here, and the next-order $\log\log r/\log r$ correction is not claimed. Companions prove a Gaussian free field limit for discrete Gaussian heights through the roughening threshold with universal value $8\pi$, and (conditional on stated inputs) the critical center magnetization $n^{-1/8}(\log n)^{1/16}$ and spin-field scaling limit.

Log-correction paper, Theorem 1.1: $C_{b_c}(r)=B_{XY}r^{-1/4}(\log r)^{1/8}(1+o(1))$ as $r\to\infty$ along the axis, with $B_{XY}\in(0,\infty)$, for the free-box limit. Exponent paper (September 24): $C_{b_c}(r)=r^{-1/4+o(1)}$. The constant is not computed; off-axis directions, other lattices and the Villain model at criticality are not covered here, and the next-order $\log\log r/\log r$ correction is not claimed. Companions prove a Gaussian free field limit for discrete Gaussian heights through the roughening threshold with universal value $8\pi$, and (conditional on stated inputs) the critical center magnetization $n^{-1/8}(\log n)^{1/16}$ and spin-field scaling limit.