  We perform a numerical study of the long range (LR) ferromagnetic Ising model
with power law decaying interactions ($J \propto r^{-d-\sigma}$) both on a
one-dimensional chain ($d=1$) and on a square lattice ($d=2$). We use advanced
cluster algorithms to avoid the critical slowing down. We first check the
validity of the relation connecting the critical behavior of the LR model with
parameters $(d,\sigma)$ to that of a short range (SR) model in an equivalent
dimension $D$. We then study the critical behavior of the $d=2$ LR model close
to the lower critical $\sigma$, uncovering that the spatial correlation
function decays with two different power laws: the effect of the subdominant
power law is much stronger than finite size effects and actually makes the
estimate of critical exponents very subtle. By including this subdominant power
law, the numerical data are consistent with the standard renormalization group
(RG) prediction by Sak, thus making not necessary (and unlikely, according to
Occam's razor) the recent proposal by Picco of having a new set of RG fixed
points, in addition to the mean-field one and the SR one.
