  We investigate the liquid state structure of the two-dimensional (2D) model
introduced by Barkan et al. [Phys. Rev. Lett. 113, 098304 (2014)], which
exhibits quasicrystalline and other unusual solid phases, focussing on the
radial distribution function $g(r)$ and its asymptotic decay $r\to\infty$. For
this particular model system, we find that as the density is increased there is
a structural crossover from damped oscillatory asymptotic decay with one
wavelength to damped oscillatory asymptotic decay with another distinct
wavelength. The ratio of these wavelengths is $\approx1.932$. Following the
locus in the phase diagram of this structural crossover leads directly to the
region where quasicrystals are found. We argue that identifying and following
such a crossover line in the phase diagram towards higher densities where the
solid phase(s) occur is a good strategy for finding quasicrystals in a wide
variety of systems. We also show how the pole analysis of the asymptotic decay
of equilibrium fluid correlations is intimately connected with the
non-equilibrium growth or decay of small amplitude density fluctuations in a
bulk fluid.
