On the merger rate of primordial black holes: effects of nearest
  neighbours distribution and clustering
  One of the seemingly strongest constraints on the fraction of dark matter in
the form of primordial black holes (PBH) of ${\cal O}$(10)$\,M_\odot$ relies on
the merger rate inferred from the binary BH merger events detected by
LIGO/Virgo. The robustness of these bounds depends however on the accuracy with
which the formation of PBH binaries in the early Universe can be described. We
revisit the standard estimate of the merger rate, focusing on a couple of key
ingredients: the spatial distribution of nearest neighbours and the initial
clustering of PBHs associated to a given primordial power spectrum. Overall, we
confirm the robustness of the results presented in the literature in the case
of a narrow mass function (which constrain the PBH fraction of dark matter to
be $f_{\rm PBH}\lesssim 0.001-0.01$). The initial clustering of PBHs might have
an effect tightening the current constraint, but only for very broad mass
functions, corresponding to wide bumps in the primordial power spectra
extending at least over a couple of decades in $k$-space.
