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The influence of non magnetic impurities in the 2d XY model is investigatedthrough Monte Carlo (MC) simulations. The general picture of the transition isfully understood from the Harris criterion which predicts that the universalityclass is unchanged, and the Berezinskii-Kosterlitz-Thouless description of thetopological transition remains valid. We nevertheless address here the questionabout the influence of dilution on the quasi-long-range order at lowtemperatures. In particular, we study the asymptotic of the pair correlationfunction and report the MC estimates for the critical exponent $\eta$ atdifferent dilutions. In the weak dilution region, our MC calculations arefurther supported by simple spin-wave-like calculations.Comment: 8 pages, 7 eps figure

Influence of quenched dilution on the quasi-long-range ordered phase of the $\mathsf{2d}$ $\mathsf{XY}$ model