  The influence of non magnetic impurities in the 2d XY model is investigated
through Monte Carlo (MC) simulations. The general picture of the transition is
fully understood from the Harris criterion which predicts that the universality
class is unchanged, and the Berezinskii-Kosterlitz-Thouless description of the
topological transition remains valid. We nevertheless address here the question
about the influence of dilution on the quasi-long-range order at low
temperatures. In particular, we study the asymptotic of the pair correlation
function and report the MC estimates for the critical exponent $\eta$ at
different dilutions. In the weak dilution region, our MC calculations are
further supported by simple spin-wave-like calculations.
