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Monte Carlo simulations of the two-dimensional XY model are performed in asquare geometry with fixed boundary conditions. Using a conformal mapping it isvery easy to deduce the exponent eta_sigma(T) of the order parametercorrelation function at any temperature in the critical phase of the model. Thetemperature behaviour of eta_sigma(T) is obtained numerically with a goodaccuracy up to the Kosterlitz-Thouless transition temperature. At very lowtemperatures, a good agreement is found with Berezinskii's harmonicapproximation. Surprisingly, we show some evidence that there are nologarithmic corrections to the behaviour of the order parameter density profile(with symmetry breaking surface fields) at the Kosterlitz-Thouless transitiontemperature.Comment: 7 pages, 2 eps figure

Correlations in the low-temperature phase of the two-dimensional XY model