  We present an analytic approach to study concurrent influence of quenched
non-magnetic site-dilution and finiteness of the lattice on the 2D XY model.
Two significant deeply connected features of this spin model are: a special
type of ordering (quasi-long-range order) below a certain temperature and a
size-dependent mean value of magnetisation in the low-temperature phase that
goes to zero (according to the Mermin-Wagner-Hohenberg theorem) in the
thermodynamic limit. We focus our attention on the asymptotic behaviour of the
spin-spin correlation function and the probability distribution of
magnetisation. The analytic approach is based on the spin-wave approximation
valid for the low-temperature regime and an expansion in the parameters which
characterise the deviation from completely homogeneous configuration of
impurities. We further support the analytic considerations by Monte Carlo
simulations performed for different concentrations of impurities and compare
analytic and MC results. We present as the main quantitative result of the work
the exponent of the spin-spin correlation function power law decay. It is non
universal depending not only on temperature as in the pure model but also on
concentration of magnetic sites. This exponent characterises also the vanishing
of magnetisation with increasing lattice size.
