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Vortex fluctuations above and below the critical Kosterlitz-Thouless (KT)transition temperature are characterized using simulations of the 2D XY model.The net winding number of vortices at a given temperature in a circle of radius$R$ is computed as a function of $R$. The average squared winding number isfound to vary linearly with the perimeter of the circle at all temperaturesabove and below $T_{KT}$, and the slope with $R$ displays a sharp peak near thespecific heat peak, decreasing then to a value at infinite temperature that isin agreement with an early theory by Dhar. We have also computed thevortex-vortex distribution functions, finding an asymptotic power-law variationin the vortex separation distance at all temperatures. In conjunction with aCoulomb-gas sum rule on the perimeter fluctuations, these can be used tosuccessfully model the start of the perimeter-slope peak in the region below$T_{KT}$.

Coulomb gas sum rules for vortex-pair fluctuations in 2D superfluids